V. S. Saf ronov
EVOLUTION OF THE
PROTOPLANETARY CLOUD
AND FORMATION OF THE
EARTH AND THE PLANETS
CASE FILE
COPY
TRANSLATED FROM RUSSIAN
Published for the National Aeronautics and Space Administration
and the National Science Foundation, Washington, D.C.
by the Israel Program for Scientific Translations
AKADEMIYA NAUK SSSR
INSTITUT FIZIKI ZEMLI IMENI O. Yu. SHMIDTA
Academy of Sciences of the USSR
Shmidt Institute of the Physics of the Earth
V.S, Safronov
EVOLUTION OF THE
PROTOPLANETARY CLOUD
AND FORMATION OF THE
EARTH AND THE PLANETS
(Evolyutsiya doplanetnogo oblaka i obrazovanie Zemlii planet)
Izdatel'stvo "Nauka,"
Moskva, 1969
Translated from Russian
Israel Program for Scientific Translations
Jerusalem 1972
TT 71-50049
NASA TT F-677
Published Pursuant to an Agreement with
THE NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
and
THE NATIONAL SCIENCE FOUNDATION, WASHINGTON, D. C.
Copyright (c) 1972
Israel Program for Scientific Translation^ I. id.
IPST Cat. No. 5979
ISBN 7065 1225 1
Translated and Edited by IPST Staff
Printed in Jerusalem by Keter Press
Binding: Wiener Hindery Ltd., Jerusalcti
Available from the
U.S. DEPARTMENT OF COMMERCE
National Technical Information Service
Springfield, Va. 22151
III/16/3
Contents
LIST OF SYMBOLS 1
INTRODUCTION 2
PART I. Evolution of the Protoplanetary Cloud and the Formation of the
Cluster of Solid Bodies 4
Chapter I. ORIGIN OF THE PROTOPLANETARY CLOUD 4
1. A few remarks on present-day theories of the origin of the
protoplanetary cloud 4
2. The angular momentum acquired by the Sun due to the
Poynting -Robertson effect H
3. Motion of solid particles in a gas driven by the magnetic
field 14
Chapter 2. TURBULENCE IN THE PROTOPLANETARY CLOUD 16
4. Condition of convective instability in rotating systems 16
5. Other possible causes leading to disruption of stability 20
6. Influence of the magnetic field on the stability of the
rotating cloud 23
Chapter 3. FORMATION OF THE DUST LAYER 2
5
7. Barometric formula for flat rotating systems 25
8. Flattening of the dust layer in a quiescent gas 26
9. Thickness of the dust layer in turbulent gas 27
Chapter 4. TEMPERATURE OF THE DUST LAYER
32
10. Statement of the problem 32
11. Temperature distribution inside the dust layer 34
12. Temperature of the layer near the surface 36
13. Warming of the layer by radiation scattered in the gaseous
component of the cloud 42
111
14. Condensation of volatile substances on particles 44
Chapter 5. GRAVITATIONAL INSTABILITY 45
15. Fundamental difficulties in the theory of gravitational
instability in infinite systems 45
16. Gravitational instability in flat systems with nonuniform
rotation 47
17. Growth of perturbations with time 52
Chapte r 6. FORMATION AND EVOLUTION OF PROTOPLANETARY DUST
CONDENSATIONS • ■ • 57
18. Mass and size of condensations formed in the dust layer .... 57
19. Evolution of dust condensations 61
Conclusions 67
PART II. Accumulation of the Earth and Planets 69
Chapter 7. VELOCITY DISPERSION IN A ROTATING SYSTEM OF
GRAVITATING BODIES WITH INELASTIC COLLISIONS ... 69
20. Velocity dispersion in a system of solid bodies of equal mass . . 69
21. Increase in energy of relative motion in encounters 77
22. Velocity dispersion of bodies moving in a gas 81
23. Velocity dispersion in a system of bodies of varying mass .... 82
Chapter 8. STUDY OF THE PROCESS OF ACCUMULATION OF PRO-
TOPLANETARY BODIES BY THE METHODS OF CO-
AGULATION THEORY 90
24. Solution of the coagulation equation for a coagulation
coefficient proportional to the sum of the masses
of the colliding bodies 90
25. Asymptotic power solutions of the coagulation equation .... 97
Chapter 9. ACCUMULATION OF PLANETS OF THE EARTH GROUP ... 105
26. Growth features of the largest bodies 105
27. Accumulation of planets of the Earth group 109
Chapter 10. ROTATION OF THE PLANETS H3
28. Critical analysis of earlier research 113
29. Methods for solving the problem 123
Chapter 11. THE INCLINATIONS OF THE AXES OF ROTATION OF THE
PLANETS 129
30. Evaluating the masses of the largest bodies falling on the
planets from the inclinations of the axes of rotation of the
planets 129
Chapter 12. GROWTH OF THE GIANT PLANETS 136
31. Duration of growth process among the giant planets 136
32. Ejection of bodies from the solar system 138
33. Dissipation of gases from the solar system 144
Chapter 13. FORMATION OF THE ASTEROIDS 146
34. Role of Jupiter in the formation of the asteroid belt 146
35. Rabe's theory of the formation of rapidly rotating asteroids . . 148
Conclusions 152
PART III. Primary Temperature of the Earth 155
Chapte r 14. INTERNAL HEAT SOURCES OF THE GROWING EARTH
AND IMPACTS OF SMALL BODIES AND PARTICLES 155
36. Warming of the Earth due to generation of heat by radio-
activity and compression 155
37. Warming of the Earth by impacts of small bodies and particles 161
Chapter 15. WARMING OF THE EARTH BY IMPACTS OF LARGE
BODIES 164
38. Thermal balance of the upper layers of the growing Earth . . . 164
39. Fundamental parameters of impact craters 166
40. Heat transfer in mixing by impact and depth distribution
of the impact energy 171
Chapter 16. PRIMARY INHOMOGENEITIES OF THE EARTH'S MANTLE . 182
41. Inhomogeneities due to differences in chemical composition
between large bodies 182
42. Inhomogeneities due to impacts of falling bodies 184
Conclusions 187
BIBLIOGRAPHY 190
SUBJECT INDEX 203
v
LIST OF SYMBOLS
G —gravitational constant;
//, h —homogeneous width and half-width of cloud (dust layer);
in Chapters 1 and 2, H is the magnetic field intensity;
A"— angular momentum;
/ —impact parameter; in Chapters 2, 3 and 15, mixing length;
m T r, o —mass, radi-us and density of growing planet;
P —period of rotation about Sun;
(? — present mass of planet;
R — distance from Sun; in Chapter 15, radius of crater;
T —temperature;
V —velocity of one body relative to another as they approach;
in Chapter 12, velocity in nonrotating system linked with Sun;
V c — Keplerian angular velocity;
v —velocity of body in system moving with angular velocity;
6 —parameter characterizing relative velocities of bodies
according to (7.12);
p —density in central plane of protoplanetary cloud;
3.V
471^3
-density obtained for uniform distribution of mass of central
body M within a sphere of radius R\
surface density of matter in zone of planet, i.e., mass in column
(of cross section/cm 2 ) perpendicular to the central plane;
Boltzmann constant;
time of free flight;
angular velocity.
INTRODUCTION
The origin of the Earth and planets is one of the most involved problems
facing science today, and it is one that can be solved only by recourse to
many disciplines. An important event in the development of planetary
cosmogony over the last twenty years has been the emergence of new
sources of data. In the forties Shmidt pointed out that the extensive data
pertaining to the Earth sciences should be useful in cosmogony. In the
fifties the rich fund of data obtained from physicochemical research on the
composition and structure of meteorites and terrestrial rocks was put to
use by Urey. In the sixties a new approach appeared— study of the nuclear
evolution of protoplanetary matter using data on the distribution and
isotopic composition of various elements in meteorites, Earth rocks, and
the Earth's atmosphere. Finally, the conquest of the planets and space in
the Earth's vicinity has begun to yield new data. Thus planetary
cosmogony rests today on a broad spectrum of data collected by related
branches of science.
In the present work we will be mainly concerned with the physico-
mechanical aspects of the emergence of the planets from the protoplanetary
material. Our discussion will be based on the concept of the accumulation
of the planets from the solid bodies and particles which, together with the
gases, made up the protoplanetary cloud enveloping the Sun in the early
phases of its development. The idea that the planets were formed by the
condensation of solid bodies and particles— advanced outside the USSR
in the 20th century by flu Ligondes, Chamberlin and Moult on — was
extensively developed in Shmidt's theory and later in the physicochemical
studies of meteorites carried out by Urey and others. This idea underlies
many geophysical and geochemical studies now being carried out both
within and outside the Soviet Union. Foremost among them is the study of
the thermal history of the Earth and planets.
In the first part of the book we will study the evolution of the gas-dust
protoplanetary cloud and the formation of a cluster of solid bodies. In the
second part we will be studying fundamental laws of dynamics in a rotating
system of gravitating bodies with inelastic collisions and the process of
accumulation of planets by the aggregation of solid bodies and particles. In
the third part we develop a method for evaluating the primary temperature
of the Earth and discuss the question of primary inhomogeneities in its
mantle.
The author is well aware that the idealized schemes considered below do
not always depict reality adequately. This is especially true for the early
stages of evolution of the protoplanetary cloud which are still very obscure
due to the absence of any definite idea on the mechanism of formation of
the cloud. However, these idealizations are an essential stage in the
development of the theory, for without them it would not be possible to
progress from general qualitative arguments to a concrete quantitative
discussion of the probable evolutionary paths. Detailed quantitative study
of the different schemes and models can establish which of them are
unsuitable and must be discarded, and which are worthwhile and in need of
further development. The main requirement to be imposed on this method
of research is that far-fetched, abstract schemes totally divorced from
reality should not be considered. In choosing the schemes, too, we
considered results and data not discussed in this work (research on the
early evolution of the stars and Sun; physicochemical, nuclear, and other
studies). The picture we draw below of the later stages in the evolution
of the protoplanetary cloud and of the formation within it of the planets is
the most probable one in our view, given the present level of our knowledge
of the solar system. Ideas about the early stages of evolution of the cloud
are still vague and subject to change. They are gradually growing more
definite as a result of theoretical studies of various cloud models. Valuable
information on conditions in the protoplanetary cloud emerges from the
study of meteorite structure. Finally, direct observation of the area
adjoining other stars is becoming possible. In 1966 Low and Smith carried
out infrared observations of the cloud near the star/? Monoceros, which
has features similar to the protoplanetary cloud enveloping our Sun. The
chief aim of this work is to give as unified a picture of this evolution as
possible in order to be able to subject all aspects to critical discussion
and thereby stimulate the development of the theory.
The author thanks Prof. B. Yu. Levin for the many critical remarks
offered while the book was being prepared.
Part I
EVOLUTION OF THE PR O TOP LAN E TAR Y
CLOUD AND THE FORMATION OF THE
CLUSTER OF SOLID BODIES
Chapter 1
ORIGIN OF THE PROTOPLANETARY CLOUD
1. A few remarks on present-day theories of the origin
of the protoplanetary cloud
Since Kant and Laplace, who first put the problem of the origin of the
solar system on a scientific footing, many works have been devoted to this
problem, and many different cosmogonic hypotheses have been advanced.
The need to make sense of the results led to the publication of a series of
extensive reviews setting forth the history of cosmogony and analyzing in
detail the better known and more interesting hypotheses. We may therefore
confine ourselves to remarks concerning contemporary hypotheses still being
debated today. Readers interested in the history of cosmogony are referred
to the reviews of Jeffreys (1952), Schatzman (1954), Spenser- Jones (1956),
Ter Haar and Cameron (1963), Ovenden (1960), Williams and Cremin (1968),
and Herczeg (1968). Also, we will not deal with McCrea's hypothesis, which
is not as yet advanced enough quantitatively.
Nowadays it is generally accepted that the planets were formed from
material rotating about the Sun in the form of an extended gas-dust cloud
filling the entire space now occupied by our solar system. The theory of the
formation of planets by a gradual accumulation of the solid particles and
bodies formed in this cloud has undergone considerable quantitative
elaboration in recent years. However, the problem of the origin of the
cloud itself has not been solved. Once Laplace's hypothesis of a common
origin of Sun and planets in a single nebula had been disproved — together
with theories postulating the formation of the planets from matter separated
from a formed Sun — it was natural to expect that the idea of capture of the
cloud by the Sun would appear in several variants. A hypothesis of gravita-
tional capture was proposed by Shmidt (1944). He based himself initially on
an erroneous scheme in which the third body participating in the Sun's
capture of material from the interstellar nebula was the center of the Milky
Way. Subsequently computations were carried out for capture in the case of
three bodies of identical mass, showing that Chazy was mistaken in stating
that capture was impossible for a positive energy constant. Evaluating the
capture probability, Shmidt (1960) was led to conclude that capture was far
more likely at an early stage in the Sun's life, before it had left the parent
medium.
Other forms of capture, not purely mechanical, were also proposed;
these did not require the proximity of another star. Agekyan (1 949, 1950) was
able to show with the aid of the accretion mechanism (Bondi and Hoyle,
1944) that the Sun could have captured dust material with the mass and
angular momentum of the planets when it was crossing the edges of the dust
cloud. Particles traveling toward the Sun under the pull of its gravity will
intersect its trajectory in a sector situated behind the Sun, setting up a
high- density region. As a result of inelastic collisions among the particles
along tliis axis their transverse velocity component is damped, leaving only
the axial component which is equal to the Sun's velocity relative to the cloud.
Particles close to the Sun, for which this velocity is less than the parabolic,
are captured by the Sun and begin to revolve around it (instead of landing
on the Sun) due to the inhomogeneity of the cloud.
Radzievskii (1950) demonstrated that it was possible for the Sun to capture
dust particles of diameter less than 10" 5 cm due to the reduced role of
radiation pressure resulting from shrinking of the particles by evaporation
as they approach the Sun. The efficiency of this capture mechanism was not
calculated. Alfven, founder of cosmical electrodynamics and magnetohydro-
dynamics, suggested capture of the cloud by the Sun with the aid of its
magnetic field (1954, 1958). However, a very high magnetic field strength
is necessary for this type of capture to be efficient. Lyttleton (1961)
surmised the accretion of gas- dust material by the Sun from interstellar
clouds of density 10~ 23 g/cm 3 and with the unrealistically low temperature of
3.18°K, rotating with the angular velocity of rotation of our Galaxy, ~10~ 15
sec" 1 . For capture of the required mass with the required angular momentum,
the relative velocity of the cloud must be ~0.2 km/sec. In the case of a
Maxwell distribution of the cloud velocities with an average of -^ 10 km/sec,
the fraction of clouds having a velocity ^0.2 km/sec is 10~ 5 . From the Sun's
birth to the formation of the planetary system (~ 1 billion years), the Sun has
encountered interstellar clouds about 10 2 times. From this Lyttleton
derives that the probability of capture of protoplanetary material by the Sun
is 10~ 3 . However, this estimate is far too high. If the peculiar velocity of
the Sun is taken as 20 km/sec, one finds that the fraction of clouds having
small velocities relative to the Sun is not 10~ 5 but smaller (10~ 7 ). Also, the
impact frequency is proportional to the relative velocity. This gives the
correction factor 0.02; instead of 10~ 3 we arrive at 10~ 7 .
Woolfson (1964) combined the capture hypothesis with the separation
theory of Jeans: the Sun came close to a star of small mass (~l/7 M j but
enormous radius (15 a. u.), and captured some of the material ejected by the
tidal swelling of the star. Calculations by Woolfson indicate that the
distance to the star at perihelion must be about three star radii. A star of
such considerable size can be regarded as in a state of gravitational
contraction. Consequently, here too a low probability is a feature of the theory.
Earlier we mentioned that the capture probability turns out to be much
larger if one assumes, in view of present-day ideas of the group formation
of stars in large clusters of interstellar clouds (Ambartsumyan, 1947;
Lebedinskii, 1954; Fowler and Hoyle, 1963), that the Sun was not born in
isolation, and if one considers capture during the period of solar genesis
when the Sun was still close to other developing stars and nebulae (Shmidt,
1957). It is not possible to state more definitely whether such capture is
actually possible before conditions near the growing Sun are investigated.
However, capture theories encounter another difficulty: they fail to
explain why the Sun's rotation and the revolution of the planets are in the
same sense. One might suppose that part of the captured material landed
on the Sun and gave it a spin in the sense of revolution of the cloud. But this
would not correspond quantitatively to the observed distribution of mass
and angular momentum in the solar system. The orbital angular momentum
of all the planets in the terrestrial group is 2 1 / 2 orders smaller than the
angular momentum of Jupiter, while the angular momentum of Mercury is
1V2 orders smaller than that of the Earth. If the captured material had had
such a monotonic variation of angular momentum with distance from the
Sun, the angular momentum of that part of the cloud nearer to the Sun than
the Mercury zone would have been no greater than the angular momentum
of Mercury itself. Therefore, if all this material were somehow to land
on the Sun, the angular momentum it would impart would not exceed
0.003% of the angular momentum of all the planets, i. e., it would have been
two orders of magnitude less than the rotational momentum of the Sun.
Shmidt (1950) and Radzievskii (1949) conjectured that the Sun could have
acquired a rotation in the same sense as the cloud's revolution owing to
solid particles landing on the Sun from the cloud due to the Poynting-
Robertson effect. Accordingly, we calculated the maximum amount of
material that could land on the Sun due to this effect (1955) assuming that
the Sun was fully formed and that its mass and luminosity were close to
what they are now. The quantity of material landing on the Sun under these
conditions due to the Foynting- Robertson effect, as found by us, could have
imparted an angular momentum equal to only 0.002 of the present rotational
momentum of the Sun. Thus capture theories fail to explain the present
rotation of the Sun.
Most astronomers persisted in adhering to classical Laplacian ideas of
a common genesis of Sun and cloud; for a long time, however, no concrete
common-genesis theory capable of explaining the fundamental laws of the
solar system was advanced. The first serious attempt was made after the
death of Shmidt. Hoyle (1960) advanced a hypothesis envisaging the common
genesis of the Sun and cloud from a single rotating nebula. Using the known
relations between the mass and radius of a contracting, rotationally
unstable protostar losing material from its equator (Schatzman, 1949;
Safronov, 1951)
d («>Mk 2 8 2 ) = mRHM, <o 2 /? 3 & GM, M = M Q (RjR^ ki ~^ ( 1 )
(where kli is the radius of gyration), Hoyle found that there is no purely
hydrodynamic mechanism capable of explaining the slow rotation of the
Sun. He then suggested that magnetic forces were the leading factor in the
transmission of rotational momentum from Sun to cloud. The solar nebula,
with initial dimensions of the order of the distance to the nearest stars
and an initial angular velocity of the order of that of the Milky Way
(-^10~ 15 sec" 1 ), was originally tied to interstellar clouds by the galactic
magnetic field. At the first stage of slow contraction, in Hoyle's view, it
transferred to these clouds the greater part of its rotational momentum.
Then, when the material became capable of moving freely across the lines
of force, the tie was broken and free gravitational contraction set in, with
conservation of angular momentum. When the nebula had contracted to the
size of Mercury's orbit it became rotationally unstable. A disk (ring) of
mass 0.01 M separated out in the equatorial region of the nebula. A strong
magnetic "bonding" set in immediately between the central condensation
(protosun), rotating as a solid body, and the inner edge of the disk
(protoplanetary cloud), so that their rotational velocities remained almost
identical and further leakage of material to the disk ceased. The substance
in the disk, having acquired angular momentum, began to move away from
the Sun and spread throughout the solar system, while the protosun, losing
angular momentum, continued to contract. The nonvolatile substances in
the disk condensed rapidly into solid particles. The particles were not
affected by the magnetic field, but they were carried off by the gas and also
spread throughout the solar system. The next process in planet genesis
consisted of aggregation of solid particles into large bodies. The energy
of rotation of the protosun must have been transformed into magnetic energy.
For this to happen, the initial field (~ 1 gauss on separation of the disk-)
must have increased in strength to 10 5 gauss, i. e., made 10 5 loops owing to
the small difference in rotational momentum remaining between the disk
and the protosun. Only a small fraction of this energy was expended in
shifting the material of the disk away from the Sun. The greater part must
have dissipated within the central condensation. It is possible that a
considerable fraction of this energy dissipated in the form of cool
(electromagnetic) activity at the surface of the protosun. The latter
maintained a level of ionization in the inner region of the disk {n + /n > 10~ 7 )
such that there was no perceptible damping of the magnetic field in this
region for the entire duration of the process under consideration (~10 years),
Hoyle' s theory was well received and achieved considerable popularity.
However, defects gradually emerged. First, according to Hoyle the
magnetic field must have transferred angular momentum only to the inner
portion of the disk. The transfer of material over large distances from the
Sun, in his view., took place by turbulence. But no one has proved that
turbulence can exist in a cloud in Kepler rotation and it is doubtful whether
it can (see Chapter 2). Such a cloud should be stable with respect to small
perturbations, and chaotic motions present from the beginning would
apparently be rapidly attenuated (Safronov and Ruskol, 1956, 1957).
Second, Hoyle was very modest in his choice of characteristics for the
cloud. The cloud mass 0,01 M s was obtained under the assumption that the
compositions of Jupiter and Saturn differ little from that of the Sun. The
estimates of other authors lead to a larger mass (see Chapter 12).
According to Hoyle, before reaching the stage of gravitational contraction
the solar nebula transferred about 99% of its initial angular momentum to
the interstellar environment. Cameron (1962) regards this as unlikely. He
adduces evidence in favor of a considerably weaker magnetic field in the
Milky Way (3 * 10~ 6 gauss) which, in his opinion, precludes significant
slowing down of the solar nebula. As a result Cameron arrives at a
completely different view of the nebula's evolution (see below).
Third, the process by which Hoyle supposes the solid material to have
been transported from the inner edge of the disk (from the distance of
Mercury) to the entire solar system— by gases — poses grave difficulties.
Hoyle links the motion of gas away from the Sun to tangential (orbital)
acceleration of the gas by the magnetic field. The orbital velocity of the gas,
in his view, must exceed the circular velocity by a quantity Av of the order
of the radial velocity v R . The particles are not influenced by the magnetic
field; they move with a circular velocity. Therefore the gas must impart
to them a tangential acceleration, under the influence of which they move
away from the Sun. Hoyle found that in effect the gas shifted away all
bodies with a cross section less than 1 m.
* Recently Hoyle and Wicramasinghe (1969) revised the initial field to 10 2 gauss.
However, for a purely tangential acceleration f v the deviation Av in the
velocity of the gas from the circular velocity is an infinitesimal of second
order with respect to f f and ^ (see Safronov, 1960b; or formula (12) below,
for f R =Q):
A 3 - i?2
Correspondingly, the size of the largest particles carried away by the gas
must be much smaller than that found by Hoyle. But the force acting on the
gas also has a radial component. According to Hoyle, as the magnetic field
twists in the disk the direction of the lines of force tends to become tangen-
tial while the direction of the force exerted by the field on the gas tends to
become radial. However, the radial acceleration f R of the gas, by weakening
the gravity of the central mass, makes the relative velocity of the gas
become smaller than the circular velocity (Whipple, 1964). Therefore the
gas does not speed up, but rather slows down the particles, making them
draw closer to the Sun. Calculations indicate (Safronov, 1966a) that the
particles move away from the Sun only at the initial stage, when they are
still small and the field is still untwisted. Their distance from the Sun can
increase only by a small fraction of an astronomical unit. Subsequently,
due to the twisting of the field and the growth of the particles themselves,
the latter begin to draw nearer to the Sun (see Section 3). This result
represents a serious stumbling-block for Hoyle's theory: if the protoplane-
tary cloud separated from the Sun at the distance of Mercury, then the solid
particles could not have traveled as far as the positions of the other planets.
To resolve this contradiction it would be necessary to revise basic assump-
tions of the theory.
A different variant of the theory of common genesis was proposed by
Cameron (1962). According to Cameron, the magnetic field was important
neither at the initial stage of evolution of the protosolar nebula, nor in its
collapse. During contraction local angular momentum is conserved and for
R >100 a.u. rotational instability sets in, leading to transfer of nearly all
the material in the nebula to the disk. After this point, and only after this
point, will the magnetic field,' which is being twisted in the disk, transmit
angular momentum outward, while the inner regions of the disk shift toward
the center to form the Sun. Cameron calculated the distribution of density
in the disk after collapse for models with an index of polytropy of 1.5 and 3
and mass 4 M e and 2 M , respectively. However, whether such an
enormous quantity of material could subsequently leave the solar system
remains unclear.
In 1963 Cameron adopted a different model for the disk presupposing that
it was formed in the contraction of a homogeneous, slowly turning proto-
stellar cluster. The surface density and angular velocity of rotation of the
disk decrease inversely as the distance from the axis of rotation, while the
mass is very nearly that of the Sun. Differential rotation intensifies the
magnetic field, due to which the greater part of the disk's mass shifts
inward to form the Sun, and only a small part, acquiring angular momentum,
moves outward. The solid matter which condenses in the latter forms the
planets. Later Cameron (1967, preprint) related the transport of material
and angular momentum in the disk to turbulence maintained by thermal
convection. At temperatures below 2000°K the disk becomes opaque and a
superadiabatic temperature gradient is established at right angles to the
plane of the disk. About half of the mass of the disk disperses and therefore
its initial mass is again taken as 2 M Q . According to Cameron, in order for
the disk to become opaque and convection to set in, its surface density must
be 10 5 — 10 6 g/cm 2 . Convection will give rise to turbulent motions not only
along the axis of rotation but also radially. Owing to turbulent viscosity
angular momentum is transmitted outward and the disk disperses. The
energetic efficiency of this mechanism is still not clear. Further, the
cessation of convection which occurs when surface density decreases due to
dispersal of the gas to 10 5 g/cm 2 makes it difficult to explain the subsequent
evolution of the disk. One should expect gravitational instability to appear
inside the disk after its temperature drops to a few hundred degrees,
leading to the formation of numerous massive gaseous condensations with
a total mass many times greater than that of the planets. It is impossible
for such a system of bodies to become our planetary system (see Section 33).
Cameron suggests that molecular absorption may have caused the disk to
remain opaque and convectively unstable for surface densities of 10 4 —
10 5 g/cm 2 . But if gravitational instability is not to prevail (in the gas) after
cooling of the disk, the surface density must be less than 10 4 g/cm 2 .
Schatzman (1962) proposed an efficient mechanism of loss of angular
momentum by the Sun involving the electromagnetic activity induced on its
surface by the interaction between the magnetic field and the convective
zone of a rotating star. The material thrown out by the star will be drawn
away by its magnetic field and will move with the angular velocity of
rotation of the star until a distance is reached where the Coriolis force
equals the pressure of the magnetic field: 2p7to~//' 2 /4ji/? c , where p and V are
the density and velocity of the ejected matter, H is the magnetic field
strength and R e is the radius of curvature of the lines of force of the magnetic
field ( /? c ~~ R is assumed). The quantity of material ejected is evaluated
under the assumption that about 10" 2 part of the magnetic energy of the
active centers is expended in ejection. The loss of matter and angular
momentum due to escape of material from the equator of the rotationally
unstable star is considered in conjunction with that caused by ejection from
the active regions. The loss caused by ejection is initially small but it
increases gradually and eventually becomes greater than the outflow.
Outflow from the equator ceases while the rotation continues to slow down
owing to the continuing ejection of matter from the active regions. The
distance of separation, and therefore the angular momentum drawn away
by the material, will be slightly smaller if the conditions for the break-up
of the bonding between the ejected material and the magnetic field are
chosen according to Cowling (1964): V = v a = H\\jh^ , where v a is the Alfven
velocity.
In 1967 Schatzman gave a more detailed exposition of his nebular theory
of the origin of the solar system. He assumes that contraction of the nebula
was due not to collapse with free fall velocities, but rather to secular
instability. The tremendous energy liberated in contraction could have left the
nebula only by convection and turbulence. A uniform rotation was main-
tained in the nebula owing to the high viscosity. We recall that in Cameron's
model opacity and convection appear only after contraction of the nebula
and the formation of the disk; during contraction viscosity is low and the
angular momentum of the material is conserved. Schatzman describes
contraction by equations (1). The outflow of material from the equator
owing to rotational instability proceeded uninterruptedly during contraction
of the nebula from the size of the Plutonian orbit to that of Mercury's orbit.
For rotationally unstable stars the values of k 2 are several times smaller
than for ordinary stars (Auer and Woolf, 196 5). For the index of polytropy
n = 3, k 2 = 0.038 and the separated mass is 0.094 M Q .
It would seem that this value for the mass of the protoplanetary cloud is
of the order of the upper limit of admissible values (see Section 32); how-
ever, its distribution over distances from the Sun is very different from the
distribution of planetary material. If to the latter one adds the light
elements required to equal the cosmic (solar) composition, it will be found
that the surface density a of material in the solar system is approximately
constant up to Jupiter, dropping off only after Jupiter as R~ z . In Schatzman' s
model occR~ 2 throughout the cloud and the cloud mass within the limits
0.3 — 3 a. u. is equal to the cloud mass within the limits 3 — 30 a. u.
On the other hand, for the isotropic turbulence adopted by Schatzman
gases behave as ideal gases with t^ 6 / 3 . But then the index of polytropy
should be «= 1.5 rather than 3, and the mass of the protoplanetary cloud
separated from the equator of the protosun would be too large (0.46 M Q ).
Schatzman belives that no less than half of the deuterium present in the
Earth and in meteorites Was formed inside the protoplanetary cloud when
helium nuclei were split by cosmic rays emitted by the active protosun.
But if this is so, then within the period when the cloud was being irradiated
by cosmic rays (~2 • 10 6 years) hydrogen must have escaped from the region
of terrestrial planets. Schatzman has shown that this is possible. It is still
not clear, however, how hydrogen could have been preserved in the zones
of Jupiter and Saturn, where the formation of massive bodies capable of
holding hydrogen would require a considerably greater interval of time.
Many aspects of the growth of the Sun and protoplanetary cloud from a
single nebula still remain vague or obscure. This is particularly true of
the role of the magnetic field in the process. Nonetheless, at this time the
idea of a common formation is more promising than that of capture. In view
of recent achievements along these lines one may hope that there will be
further progress toward a solution of the problem of the origin of the proto-
planetary cloud, in close conjunction, of course, with advances in stellar
cosmogony. As it happens, the theory of planet formation in the proto-
planetary cloud by the accumulation of solid material antedates first
attempts to broach the question of the origin of the cloud itself. By establish-
ing the distinction between these two branches of planetary cosmogony
Shmidt advanced work on the theory by fifteen years. At the time, indeed,
the fundamental laws governing the process of stellar formation were
obscure. Data on the Earth's chemical composition (deficit in inert gases,
etc.) and on the composition and structure of meteorites pointed to the
formation of the planets out of solid material. Planet formation depends
directly on the problem of the cloud's origin only at the earliest stage, that
of formation of solid bodies from diffuse material. The second stage, the
aggregation of solid bodies into planets, displays its own typical laws which
to a large extent efface the previous evolution of the cloud. Investigation of
these laws has made it possible to draw a picture of planetary formation
and relate it to the subsequent state of the planets.
An investigation of the early evolution of the cloud must rest on concrete
premises regarding its origin. The cloud model we will consider is closest
to that adopted in Schatzman' s theory. The initial mass of the cloud is about
10
0.05 M Q . Solid particles are not carried away by gaseous flows on the
periphery of the solar system as assumed in Hoyle's theory. The contraction
of the protosun and its active stage lasted about 10 7 years (Fowler and Hoyle,
1963; Ezer and Cameron, 1965). The interval of cloud formation (contrac-
tion to dimensions of Mercury's orbit), as well as the period of formation
of the protoplanetary bodies within the cloud, was considerably shorter.
Bodies of asteroidal size developed within 10 7 years; irradiation of small
bodies and particles by the active protosun had a decisive influence on their
chemical and nuclear evolution. On the other hand, the rate of the cloud's
evolution and the rate of accumulation of the bodies vary with distance from
the Sun. When discussing the cloud's evolution below we shall therefore
have in mind the sequence of events at a definite distance from the Sun, and
not events unfolding simultaneously throughout the cloud.
2. The angular momentum acquired by the Sun due to the
Poynting-Robertson effect
We saw in Section 1 that theories involving capture of the protoplanetary
cloud by the Sun fail to explain why the Sun rotates in the same sense as the
planets around it. It has been suggested that the Sun's rotation is due to
material landing on it from the cloud owing to the Poynting-Robertson effect.
We now proceed to evaluate this effect, demonstrating its inadequacy
(Safronov, 1955).
Particles moving around the Sun are slowed down by its radiation; their
orbits shrink and they slowly draw closer to the Sun. This phenomenon,
known as the Poynting-Robertson effect, is quite simple to describe if one
considers an isolated particle being acted upon only by the forces of the Sun's
gravity and radiation. In the protoplanetary cloud the particles lay in a gas
and their motion depended essentially on the motion and density of the gas.
If the only force acting on the gas was that of the Sun's gravity, it must
have moved with a Keplerian angular momentum greater than the circular
velocity of the particles which were subjected to the pressure of the Sun's
radiation. The gas accelerated the particles in the direction of their motion
and they drew farther from the Sun. But given the twisting of the Sun's
magnetic field, the circular velocity of the gas must have been less and the
particles must have drawn nearer to the Sun (see Section 3). After the gas
dispersed, the particles also moved toward the Sun. We do not understand
conditions in the cloud well enough to make definite statements about the
particles' motion. Once they reached a distance of 0.03 a. u. from the Sun,
the solid particles evaporated. Having absorbed light quanta and experienced
particle collisions, the molecules lost part of their orbital angular momen-
tum and came close to the Sun. It seems the corpuscular impacts were more
efficient. But in this process some of the molecules acquired high velocities
and left the area of the Sun. It is still not clear what fraction of the
molecules landed on the Sun's surface. However, the fact that the actual
efficiency of the mechanism under consideration is indeterminate does not
prevent one from evaluating the upper limit of the angular momentum which
the Sun could have acquired due to the Poynting-Robertson effect. The flow
of matter to the Sun was determined only by the quantity of solar radiation
trapped in the cloud. Consider two simple schemes: the motion of isolated
11
particles reemitting solar radiation, and the combined motion of the
particles and gas which trap incoming solar corpuscles.
A particle of mass m moving around the Sun along a circular orbit of
radius R, upon isotropic reemission of the incident light of mass dm r , will
lose the angular momentum \jGMRdm r , where M is the mass of the Sun. The
angular momentum imparted to the particle by radiation is negligible. With
reduction of its angular momentum the radius of the particle's orbit will
decrease:
md sjGMR = ~sjGMRdm r .
(2)
Integrating (2) we obtain the reduction of the orbital radius of the particle m
due to the radiation of mass m r incident upon it:
R^R n e
(3)
Corpuscles trapped in the cloud produce a slightly different effect.
Assuming that the particles move together with the gas, let us consider the
motion of a certain elementary volume of the cloud around the Sun. As in
the previous instance, the distance of this volume from the Sun is determined
by the law of conservation of angular momentum. For a volume with initial
mass m and initial circular orbit of radius R , disregarding the angular
momentum of the corpuscles, we obtain
(m -j- m t ) \JGMR = m \JGMR Q
(4)
and
*«(*£=:?*.
(5)
where m c is the mass of corpuscular emission trapped in this volume.
Table 1 gives the values of mjm and mjm for R equal to the distances of
various planets from the Sun and R equal to the Sun's radius.
TABLE 1
Mercury
Venus
Earth
Mars
Jupiter
mjm ....
mjm ....
2.2
8
2.5
11,5
2.7
14
2,9
17
3.5
32
Of the entire solar emission of mass AM, the cloud absorbs the mass
QAM/4n, where Q is the solid angle subtended at the Sun by the opaque
portion of the cloud. This causes the following mass of material to land on
the Sun:
Am = aj -T£- AM = ctjjAA/.
(6)
12
In the case of photon emission a x is equal to the ratio m/m rl the inverse of
which is given in Table 1. For the inner edge of the protoplanetary cloud
one can assume m/m r ^I 2 . In the case of corpuscular emission, the Sun
receives not only the cloud material m but also the trapped corpuscular
emission of mass m e$ which is one order greater. For a x we obtain the value
-~ 1 1 Thus for photon emission a 1 ^ 1 / 2 and for corpuscular
1.1.
emission a^ 1,
Since not all corpuscles landing in the cloud are trapped inside it, it
seems that in the second case a x is less than unity and not very different
from its value for photon emission. For a relatively flat cloud having
thickness H at distance R from the Sun, Qi4n&H/2R. For the usually accepted
value H& 1/25 R and <x 1 = 1 / 2 , we have a 2 = 0.01. The mass Am reaching the
Sun from the cloud in 10 8 and 10 9 years due to solar photon emission— at the
present rate (total emission = 4 • 10 12 g/sec = 1.2 • 10 29 g in one billion years)
— is given in Table 2 for s everal values of a 2 . The corresponding angular
momentum &K~bm\jGMRQ imparted by this material is given in the second
row, in terms of the present angular momentum of the Sun K e . For a 2 = 0.01,
about 0.1 of the mass of the inner planets lands on the Sun from the cloud in
10 9 years. The angular momentum imparted to the Sun by this material
amounts to only 0.002 of the present solar angular momentum. For the Sun
to have acquired its present rotation, the added mass should have amounted
to 10 2 Earth masses. Therefore, the Poynting- Robertson effect could have
caused the present rotation of the Sun only for a solar radiation L two to
three orders of magnitude greater than the present value (see last row of
Table 2).
TABLE 2
Q
a 2=°147
0.001
0.01
0.1
A*, years , .
Am, g . . .
l/L Q • • •
10 8
10 25
2- 10" 5
6 • 10 4
IO 9
10 26
2 ' 10" 4
8 • IO 3
io 8
!0 26
2 -IO" 4
6 -10 3
TO 9
!0 27
2 -IO" 3
6 -10 2
10 8
IO 27
2 ■ ID'*
6 * 10 2
:0"
10 28
2 • 10" 2
6 * 10 1
The Sun's luminosity may have been two orders of magnitude greater
than today in the closing stage of gravitational contraction (Hayashi, 1962).
But the duration of this stage did not exceed IO 7 years, and the total amount
of radiation lost by the Sun in this period did not exceed the radiation lost
by the Sun in IO 9 years. Consequently, even if the Sun's capture of the cloud
began before the onset of the high- luminosity phase, when the Sun was still
newly formed, the Poynting- Robertson effect could not have imparted the
required rotational angular momentum.
Vinogradova (1961) has shown that the rotation of the particles makes it
necessary to allow for the anisotropy reemission of solar radiation by them.
A particle rotating in the same direction as the Sun will emit less in the
direction of motion along its orbit than in the opposite direction. In the
process it will acquire positive angular momentum and move away from the
Sun. A particle rotating in the opposite direction will draw closer to the Sun.
The effect is maximum for a certain velocity of rotation determined by the
13
dimensions and physical properties of the particle. In this case it is stronger
by three orders of magnitude than the Poynting- Robertson effect. Assuming
maximum efficiency, this mechanism could have caused a substantial
redistribution of angular momentum in the solar system, as suggested by
Vinogradova, and could also have given the Sun its present rotation. But in
reality its efficiency must have been considerably below the maximum,
first because the rotational velocities of the particles vary (only in very
rare cases do they approach values for which the effect is maximum), and
second because particle collisions caused the magnitude and direction of
the rotational velocities to alter sharply, direct rotation being replaced by
inverse rotation and vice versa. Thus the motion of particles along R was
nondirectional— of the nature of random flight, or diffusion— and the
distance of a particle from its initial position increased not proportional to
the time t but to v^*
3. Motion of solid particles in a gas driven by the
magnetic field
We assume that in the absence of a magnetic field in the protoplanetary
cloud every volume element of the cloud travels around the Sun along a
circular orbit with a Kepler velocity. From the entire magneto- hydro-
dynamic problem, we will consider only the one-sided action of the magnetic
field on the motion of the gas. Suppose that this effect is expressed in the
presence of the radial and tangential accelerations f R and f v of the gas.
Disregarding the gas pressure gradient and viscosity, we can write the
equations of axisymmetric motion (Landau and Lif shits, 1953) as
to*
dt
dv R
dR
'* dR
R ~~ /?2 "T/ff
9 -h B — r r
(7)
(8)
For small f R and f 9 the motion can be treated as stationary and nearly
circular. Then
From (9) we obtain
Vr dvy v 9 w 2&v 2 dbv
~dR~ + ~R L + ». + " dR
Equations (7) and (8) yield
foR , GM
v\ = V\ + 2V£v + (An)« = =Uf R--£ + Zg— Rf R
(9)
(10)
(11)
14
and
dv,
R
^=-l=- + w *Tfr "Wr- (12)
Retaining only first- order terms, we have
*,«^.. *,*-£. (13 )
Thus Ay is practically always negative. Under the influence of the magnetic
field the gas moves around the Sun with a velocity less than the circular
Kepler velocity. Only for /*■</,, is At>> 0, having the form (7) and being a
second- order infinitesimal.
The motion of dust particles can be determined from the same arguments,
except that / and /* must be replaced in (13) by the perturbing accelerations
g^ and g R acting on the particle from the gas:
ff t = C«(Ai;-Ai;,), *« = C* (w, — i; M ) f (14)
where C = 2o f /Tcr5; r and o are the particle radius and density, o g is the surface
density of the gas, and v pR and Au p are the components of the particle velocity.
Therefore
"«,*» — = 2C (Aw — A«); Ai>„«* — 4^-= — — (v —v \
pa w \ p /t p 2ut 2 v R PR)
and finally
^=(T^K-7> ^ = ^(t + t)' (15)
Small particles for which C 2 >1 and 2Cf^f R practically move together with
the gas. Particles for which 2C/ f </ ft travel to the Sun. The corresponding
condition for the particle size has the form
'>r = -^ct g e, (16)
where is the angle between the radius vector and the line of force.
Directly after separation of the disk the magnetic field is still untwisted;
ctg 9^-1 and for o^^-lO 3 the value of r is of the order of several meters. In
practice all the particles are driven off by the gas. However, within a few
tens of years twisting of the field causes r to shrink to a few centimeters.
Within this time many particles grow to the radius r> r and begin to draw
closer to the Sun, having traveled only a small fraction of an astronomical
unit away from it. As the field twists its intensity increases. In the process,
6 -►n/2and ctg 6^ n _1 , where n is the number of loops. When, according to
Hoyle, n reaches 10 5 the critical radius r will be less than 10~ 2 cm;
practically all particles will begin to move under the influence of the gas
toward the Sun, and not away from it, as supposed by Hoyle.
Thus, it seems that the transfer of solid particles by the gas, assumed
in Hoyle ! s theory to have taken place from the Mercury zone to the orbits of
the other planets, is not possible.
15
Chapter 2
TURBULENCE IN THE PROTOPLANETARY CLOUD
4. Condition of convective instability in rotating systems
The problem of turbulence, or more precisely of the chaotic macroscopic
motions which may have appeared in the protoplanetary cloud during its
formation, is vital for understanding the cloud's evolution. The character
of subsequent processes in the cloud must have depended on whether these
primordial motions were damped or whether some kind of stationary
turbulence was established in the cloud material. The persistence of
turbulence would have prevented the separation of dust and gaseous compo-
nents and the formation of dust condensations. Planetary nuclei could then
have been formed only by the direct growth of solid particles due to agglom-
eration in collisions.
The idea of turbulence was introduced to cosmogony by von Weizsacker,
by way of a return to Descartes' classical eddies (1944). Von Weizsacker
pointed out that the Reynolds number for a cosmic diffuse medium is very
large, far above the critical value. Since then turbulence has been regarded
as one of the most widespread states of cosmic matter. Von Weizsacker
also conjectured that turbulence was important in the formation of the
heavenly bodies and their systems as well. In his view the planets, stars,
Milky Way and other structures were formed from turbulent eddies of
eration in collisions.
For the protoplanetary cloud the Reynolds number Re = — - was greater
than 10 10 . To explain the law of planetary distances, von Weizsacker
assumed that the turbulent motions within the cloud constituted a regular
system of eddies whose sizes were proportional to distance from the Sun.
Without going into a detailed analysis of all the tenets of von Weizsacker' s
theory, we consider only the fundamental idea of the prolonged persistence
(-^2 ■ 10 8 years) of turbulence in the protoplanetary cloud.
In rotating systems the Reynolds number is not the paramount criterion
of stability of motion. If one wishes to understand the motion of matter in
the protoplanetary cloud, which was a fairly flat system, one can turn to
results of studies of the motion of a fluid between two rotating cylinders
(Couette flow). According to Rayleigh's well-known criterion for incompres-
sible inviscid fluids (1916), the necessary and sufficient condition for
stability of a purely rotational motion with angular velocity o> (R) is
^<"* 2 ) 2 >0 (1)
16
throughout the fluid under consideration. Instability arises if this condition
is violated anywhere. Rayleigh's criterion was confirmed in theoretical
and experimental studies by Taylor (1923). It was found that a liquid's
viscosity increases its stability. Chandrasekhar pointed out that while the
fluid is necessarily stable if Rayleigh's criterion (1) is met, it is not
necessarily unstable if Rayleigh's criterion is not met (1958). The same
author carried out a theoretical analysis of the problem of stability for the
more general case where the distance between cylinders is not small
compared with the radius. The calculations for RJR l ^ 2 were confirmed
by experiments of Donnelly and Fultz (1958, 1960).
According to Rayleigh's criterion (1), the protoplanetary cloud should be
stable. For circular Kepler motion the angular momentum is proportional
to \/R, i. e., increases with R, and the stability condition (1) is met. The gas
pressure within the cloud is low, and the motion should be almost Keplerian.
Disregarding the gas pressure gradient dp/dR, condition (1) applied to the
flat protoplanetary cloud reduces to the condition of stability for circular
orbits known from stellar dynamics (see Chandrasekhar, 1942):
M* 1
ft)>o.
where <D is the potential energy at the distance R from the axis of rotation
(and axis of symmetry) of the system. The mass of the cloud is small
compared with the solar mass, and gravitation is determined predominantly
by the central body, i. e., d®/dR=GM/R*. Thus condition (2) is met and
therefore circular orbits in the protoplanetary cloud are stable.
However, conditions (1) and (2) fail to allow for the possibility of
convection appearing in the cloud. Von Weizsacker attempts to substantiate
the persistence of turbulence in revolving cosmic gaseous masses, including
the protoplanetary cloud, by means of the condition for the appearance of
convection (1948). But he disregards the rotation and does not take the
stability condition (l) into account. Concurrent analysis of these conditions
led to the following results (Safronov and Ruskol, 1956, 1957).
Suppose a liquid is in a purely rotational laminar motion o> {R), where R
is the distance from the axis of rotation. The sum of the forces (gravitational,
centrifugal and pressure) acting radially upon any element of the liquid is
zero:
/— t£ + »*-H& = «- (3 )
Now if we apply to this element the small perturbation &R, conserving its
angular momentum wi? 2 , it will be acted upon by the following force per unit
mass:
8/ = 2|* M _3aW*_(J-£Y +!&. (4)
' ft 3 \ p e i d/t/R+ZR ' p dR \^}
The subscript "el" indicates that the given characteristic refers to the
element under consideration. The motion is stable if the direction of the
force 6/ is opposite to the direction of displacement, i. e., 6/ < for bR >0.
Since (3) is satisfied for all R, its derivative along R is zero:
17
/^-tW + ^-W-Ct^U + T*- - (5)
Subtracting (5) from (4), we obtain
8/=-[^+ (u ,« J Rn^-^.( F L_i-) s+js . (6)
At the distance R, i.e., for an unperturbed element, P e j = p. Therefore
&-TL.=--k[&lr#\ w - (7)
Furthermore,
B** + (v*Ry = 2 %■{*&)'. (8)
Therefore the condition for stability of rotational motions with respect
to convection S//S/?<0 has the form
£«**>$■&[(&)«-&]• (9)
When the right-hand side is zero this condition reduces to Rayleigh's
criterion (1), and when the left-hand side is zero to the ordinary condition
for the onset of convection in a nonrotating medium.
If the element of volume is moving adiabatically (the adiabatic index
T = V c t)> then
/ d?\ ___Li__^£_ J*p___P__<*p LlL. (10)
\dR )u ~ t p dR * dR p dR T dR
and
\dR/*d dR—tTldR w ^ p dflj' V '
The condition for the onset of convection has the form*
(12)
The right-hand side represents the adiabatic gradient in a revolving medium.
For small displacements hR the smoothly varying functions p, p and T
can be approximated by power functions
P~i?— •; p~R-+ ; T~R—* 9 (13)
* This condition could have been expressed in the more usual form jd~< — Tds" ~Ir~ (*****)* > wnere
GM '
R*
18
where a s = « 2 — a,. Then, taking vK* from (3), we can reduce the convection
condition (12) to the simple expression
GMy.
«.>(T-l)«i + -^|g||r+T(2— ,)
2a 3 >i + ( 1 -T- 1 )a 2 + 2, < 14 >
where I ~ ^^l ; 9t is the gas constant and ju the molecular weight.
The minimum £=£ in the right-hand side of (14) w ; .ll be obtained if we
assume that the protoplanetary cloud is transparent and that its temperature
is that of a black sphere at the corresponding distance from the Sun. Then
for the present solar luminosity
3 00°
"a. u.
is the distance in astronomical units and
r.«:P=. (15)
6 ^ 3-6. lpy _ 108 / 16 j
Clearly, then, the condition for convection (14) could not have been met
in any significant part of the protoplanetary cloud. The coefficient a % , which
is related to the pressure gradient, cannot be large. Therefore from (14)
convection would require a very high temperature gradient (a a ^g). This is
excluded in a transparent cloud since from (15) a i = I / l . A high temperature
gradient could exist along the innermost edge of an opaque cloud provided
the cloud boundary was sharp. The width of the resulting zone would
obviously be very small [~Rla z <10" 2 a,u.). Convection in such a narrow
zone could not have had a perceptible effect on the cloud's dynamics.
However it might have increased the width of the inner edge, causing
screening of solar radiation by the solid particles, a drop in the cloud
temperature, and therefore alteration of the chemical composition of the
particles (see Chapter 4). However, a sharp inner boundary is ruled out by
the Poynting- Robertson effect (see Chapter 1). According to Fesenkov
(1947), in transparent solar space the density of a stationary stream of
particles of uniform size moving under the influence of this effect toward
the Sun varies as R' 1 . But along the edge of an opaque cloud the density of
the dust component must have decreased toward the Sun owing to two
factors: the radiation density increased faster than R^ due to the falling off
of absorption, and the particles shrank by evaporation. Therefore the
temperature gradient necessary for convection could not have become
established in this band either.
Thus the revolving protoplanetary cloud was stable with respect to small
perturbations, convection could not have set in inside it, and von
Weizsacker's conjecture regarding turbulence due to convection has not been
confirmed.
19
5. Other possible causes leading to disruption of stability
The formation of the solar cloud was not a smooth process, and primor-
dially the cloud may have contained random macroscopic motions of large
scale. The cloud's stability with respect to small perturbations, demon-
strated above, does not necessarily imply that these random motions were
damped quickly. * Therefore the question of the possible persistence of
"turbulence" in the cloud needs to be discussed further.
If the energy of the primordial random (i. e., with respect to the circular
Kepler velocity) motions was not dissipated, these motions would persist for
an indefinite time. This is the property with which von Weizsacker endowed
his ordered system of eddies. All the particles in an eddy move along
Kepler ellipses with the same period and without loss of energy; the center
of the eddy moves along a circular orbit. It would have been more natural
to suppose that the centers of the eddies move along elliptical orbits. But
then there would inevitably have been dissipation of the energy of relative
motion — rounding out of the orbits during motion in a resisting medium and
consequent damping of turbulence.
We could assume, in common with von Weizsacker (1944), that the
energy source which maintains turbulence inside the cloud is the gravita-
tional energy of the cloud in the Sun's gravitational field, liberated as the
inner regions of the cloud approach the Sun. Von Weizsacker (1948) and
Lust (1952) describe the evolution of the revolving turbulent cloud by means
of the ordinary equations of hydrodynamics, merely replacing molecular by
turbulent viscosity T] = p/i>. For the large-scale turbulence they assume the
value of T] is large and the equations predict a highly efficient transport of
material (from outer regions of the cloud outward and from inner regions
sunward)."- This purely formal application of hydrodynamics to turbulent
motion, however, is not justifiable. The mixing length I is comparable
with the dimensions of the system (z«0.6i?) and the velocity distribution is
not Maxwellian. The authors take shearing stresses dependent, as usual, on
the angular velocity gradient:
In Prandtl's semiempirical theory of turbulence the shearing stresses are
assumed to depend on the gradient of angular momentum:
The above relation is also used by von Karman (1953). This distinction is
very important for the evolution of the protoplanetary cloud since angular
velocity inside it decreases away from the Sun while angular momentum
increases, i.e., transport takes place in opposite directions depending on
the point of view. Taylor's experiments (1923) tend to favor the Prandtl-
von Karman view, although the Prandtl theory is excessively simplified and
Here too an analogy can be drawn with the motion of fluids between two revolving cylinders. Experiments
show that there exists a region of fairly large Re in which stationary motion is metastable: it is stable witli
respect to small perturbations but unstable with respect to large ones (Landau and Lifshits, 1953).
Analysis of the energies involved (Tei Haar, 1950) has shown that the decay time of such a cloud
(10 — 10' 1 years) is five orders of magnitude less than the time required (according to von Weizsacker) for
the planets' growth.
20
the real picture of turbulent motion is far more complex. Vasyutinskii
(1946) has put forward a more general expression for the shearing stresses
in a revolving medium:
"^^TR^)- 2 f K > (19)
where K% and K* characterize the mean transfer vl in the radial and trans-
versal directions. For isotropic transfer (#* = /£*) it reduces to the ordinary
hydrodynamic expression (17) and for purely radial transfer (K^ = 0) to the
expression (18) of Prandtl. For the protoplanetary cloud ( co — R-'t* )
1 R
Vasyutinskii' s relations lead to damping of turbulence when A"? < ^ /£* ;
however, they do not make it possible to evaluate either the ratio KI/Kr or the
scale of the turbulence. Moreover it is not clear to what extent these
generalizations are physically justified. However, Vasyutinskii's conjecture
that expressions (17) and (18) for o R correspond to two extreme cases and
that for real turbulent motion o^ should be describable by an intermediate
relation appears to be reasonable.
Using a' R it is possible to estimate how much energy of ordered rotational
motion (replenished in turn by potential energy in the gravitational field of
the central mass) is converted by viscosity into energy of random motion.
Expression (17) gives us the amount of energy converted into heat per cm 3
per second for laminar rotational motion of a fluid (Lamb, 1932):
*=,*(£)'. (20)
where tj— i-pyX and the mean free path \<^R, Similarly, expression (18)
deriving from Prandtl' s theory gives the amount of energy converted into
turbulent motion:
E
= £[^J, (21)
where Tj = -g-pyJ is the turbulent viscosity, / the mixing length, and v the
turbulent velocity. For Kepler rotation (u>~ R-*t*) the above expression
differs from (20) only by the factor 1 / 9 . Assuming that the correct value of E
for turbulent motion lies between (20) and (21), we can take expression (20)
as our basis and introduce the factor V 9 <?'< 1 in the right-hand side. In
Chapter 7 this method is used to estimate the velocity dispersion in a
system of gravitating bodies with Kepler rotation, since for large free paths
the nature of the transfer should be the same as in turbulent motion. The
value p'^0.2 was obtained.
Thus the turbulent energy acquired per unit mass per second can be
taken as e' = p'i?/p. The turbulent energy which dissipates (converting into
thermal energy) is s'^j-^-jr P er gram per second in well-developed
turbulence. By comparing e' with e* it is possible to determine whether the
turbulence is being enhanced or, on the contrary, damped:
. = «'-«* = ^^ i - 1 )=^(T^^- 1 )- <«>
where t is the mean free time of the eddies and P is the formation period.
21
Expression (22) holds only for x<^P. In this case e<0. Consequently
such small-scale turbulence must die down. For large t, I 2 must be replaced
by V 2 A ^ 2 > where Ai? is the change in the distance R of the eddy during time t.
Let us assume that one-third of the eddies has only a radial component of
relative velocity v~v R , one- third has v — v 9 , and one third has v = v M . For
eddies with v~v M , A/? = 0. For eddies with v = v R and v — v^we will take, in
accordance with (7.24)* and (7.25),
&R R = ±-Ri e \ ABj = i/?V. (23)
Since the first term of (22) already contains a factor V 3 in the expression
for 7j, the sum Sfl| + Sff2 should be used for IW 2 . We take v 2 = ^e*V\ in
accordance with (7.28). Then for large x
The value p' = 0.2 obtained in Chapter 7 yields e< 0. Thus the turbulence
must subside. For turbulence to persist it is necessary that p'>4> which
is apparently unrealistic.
In the above we have taken e*=v s /2Z under the assumption that the energy
v*/2 is dissipated within the mixing time t = lfv, and that t\z=^-pvl by analogy
with molecular viscosity. If we were to take the usual value e"ttv*/l, we
would have to conclude that turbulence dies out for any possible p\ But if
we further assume that r\&pvl , attenuation of turbulence would occur only
for p' <V,. The value fT = 0.2 satisfies this condition as well, but with a
comparatively small margin. Unfortunately, owing to the fact that fundamen-
tal relations of the theory of turbulence (which are defined only for a
constant factor) were used, conclusions regarding the attenuation of
turbulence in the protoplanetary cloud can be neither rigorous enough nor
final. All one can say is that the foregoing argument tends to favor attenua-
tion over persistence of turbulence.
It should be pointed out that the stability of the protoplanetary cloud is
deduced on the assumption that angular momentum increases away from the
center within the gravitational field of the central body, and that the stability
condition (1) is met. Strictly speaking, however, one should allow for the
gravity of the cloud as well. From stellar dynamics it is known
(Chandrasekhar, 1942) that, in the equatorial plane of a homogeneous, highly
compressed spheroid, the force of gravity near its edges (on the outside)
decreases faster than R~ 8 , and that circular orbits are unstable if the
eccentricity e of the spheroid's meridional cross section exceeds the critical
value <?!= 0.834. A similar result will be obtained in the presence of another
central mass, except that e x will be larger. If the central mass is ten times
larger than the mass of the spheroid ( a permissible assumption in the case
of the Sun and protoplanetary cloud), then e x m 0.985. But the flattening of the
protoplanetary cloud was even more pronounced. The ratio of the semiaxes
of the spheroid c/ a can be assumed to be roughly the same as that of the
* When referring to formulas from another chapter, the number of the chapter will be indicated by the first
figure and that of the formula by the second.
22
half- thickness of cloud to its distance from the Sun, which is taken to be about
1/30 for the gaseous component of the cloud. Then 1 — e*=c 2 /a* & 10" and
#^0.999>e 1 . However, in order for a region of instability to exist near such
a strongly compressed, nearly homogeneous spheroid, the gradient of
density in this region ought to be very high. The instability criterion
associated with high gradients in flattened rotating systems was given by
Lindblad in the form (see Chandrasekhar, 1942)
P-P>2^. (25)
where
a _ 1 (**\
But this criterion does not give the condition for the density gradient in
explicit form. The order of magnitude of the required gradient can be
estimated as follows. Let a nearly constant density distribution be replaced
at distance R from the center of the system by a sharply decreasing law
p— CR~ n . Then Ap/p— — nAR/R. For a homogeneous spheroid at whose boundary
the density drops abruptly to zero, there will exist a zone of instability near
its edge extending from a (maximum radius) to ae/e lt i. e., having width
Aa=(ele 1 — \)a. For the instability to be maintained in this zone with a gradual
decrease in density, it is necessary at the very least that p drop to zero
within the zone, i.e., Ap~ p . Taking Ai? = Aa, we find that
"~AF=r^T- (26)
For e= 0.999 and e x = 0.985, one obtains n» 70. In the zone of the giant
planets, the density determined from present planet masses decreases
approximately as R' s . The density gradient necessary for instability is
unattainable in any part in the protoplanetary cloud.
6. Influence of the magnetic field on the stability of the
rotating cloud
Some idea of the magnetic field's influence on the stability of the rotating
cloud can be arrived at from certain results of Chandrasekhar (1961) for the
motion of fluids between revolving cylinders (Couette flow) in the cases of a
magnetic field H s parallel to the axis of rotation and H v along the direction
of rotation. For a field H M of infinite conductivity, the stability condition is
found to be
This result is somewhat unexpected, since the above does not reduce to
Rayleigh's criterion when H -+ 0. For a vanishingly small field when w is a
monotonic function of R, the necessary and sufficient condition for instability
is that© increase with R, In the protoplanetary cloud to decreases with R
23
and therefore for a weak magnetic field the cloud should be less stable than
we found earlier in the absence of the field, when for its stability it was
sufficient that wi? 2 increase. However, this result is not confirmed when
the premises are more general and allowance is made for the dissipative
properties of the medium— the viscosity v is not zero and the electrical
conductivity a is not infinite. In the case of the magnetic field #, lying along
the axis of rotation, taking the distance between the inner and outer walls
of the cylinder {d=R 2 ~ i?J to be small compared with R and writing the
angular velocity of rotation in the form (a=A-\-B/R 2 , Chandrasekhar obtained
a theoretical expression for the dependence of Taylor's critical number T e
on the dimensionless parameter Q — jj. # 2 d 2 /4irpv?i , where tj — l/4nji t a; \i+ is the
magnetic permeability.
The dependence of t 9 on Q is almost linear. For Q -> oo(a ->ooorv ->0)
the ratio TJQ tends asymptotically to the constant value #(107 for nonconduc-
ting, 451 for conducting walls). Since
T = H\ + V )A-S£-, A = " 2 f l Z'2f' ' (28)
the stability condition T < T c can be written as
^ > ^N(R\-R\) * [ ^ }
Hence for H -*0 the stability condition reduces to Rayleigh's criterion:
increase of &R 2 with R. The presence of a magnetic field increases
stability.
In a system with differential rotation, the toroidal field H f is more
probable. Then the stability condition for a=oo and v= has the form
_d_
dR
W-^*1F(£) ! > - (3°)
For H 9 -+0 this condition reduces to Rayleigh's criterion. If //^ increases
more slowly than R, the magnetic field will increase the stability of the
rotating cloud. But even a field rapidly increasing with R would be unable
to neutralize the stabilizing effect of rotation if its strength inside the cloud
was less than 10 oersted, and the cloud would continue to remain stable.
The protoplanetary cloud probably had a toroidal field which grew weaker
away from the Sun. The presence of such a magnetic field could only have
contributed to the stability of the cloud as obtained above without allowance
for a field.
24
Chapter 3
FORMATION OF THE DUST LAYER
7. Barometric formula for flat rotating systems
We shall say that a system is flat if its thickness H is much less than the
distance R from the center of the system. The protoplanetary cloud belongs
to this category. Its thickness is determined by the thermal velocities of the
particles and can be obtained from an expression similar to the barometric
formula for the Earth's atmosphere. Let us assume that the cloud consists
of a one-component gas in laminar rotation. Its equilibrium in the radial
direction (perpendicular to the axis of rotation) will be maintained mainly by
the rotation. The gas pressure gradient along R will be very low (von
Weizsacker, 1944), and the rotation almost Keplerian, i. e., the force of
gravity is balanced by the centrifugal force. By contrast, equilibrium in the
z direction (perpendicular to the central plane) is maintained by the pressure
gradient
*-* (i)
where Z is the acceleration of gravity in the z direction. It is due to the
Sun's and the cloud's gravitation. The latter is not important and can be
disregarded provided the cloud's density is several times less than the
critical value for which gravitational instability appears inside the cloud
(see Section 16). Then
2 GMqz
(/f2 + r J)V.
GMqz
i?3
where R is the distance from the axis of rotation. Assuming that the mean
velocities of the particles do not depend on z (identical particles and constant
temperature) we obtain
? dx 3 p dx
and
L*S.t?±*L = _j t (3)
3co» *»
PW = PoP 2 u, « (4)
Consequently, the thickness H of the homogeneous layer is
25
where we take v 2 =-£-v 2 , which holds for a Maxwellian velocity distribution.
8. Flattening of the dust layer in a quiescent gas
In Chapter 2 we saw that chaotic macroscopic motions arising during, the
formation of the gas-dust cloud enveloping the Sun were rapidly damped and
that the rotation of the cloud tended to become laminar. The fact that the
chemical composition of the planets differs from that of the Sun (i. e., from
the assumed primordial composition of the cloud) indicates that the density
of the gaseous component of the cloud was not so high as to lead to gravita-
tional instability and the resulting formation of stable gaseous clusters.
The cloud's subsequent evolution must therefore have been linked to the
presence of a dust component.
Once the turbulent motions in the gas had been damped, solid particles
began to settle on the central plane. The settling time can be estimated
from the equation for the motion of a particle along the z axis. For constant
particle mass it has the form (Safronov, Ruskol, 1957)
<*. + t ,*+* = 0. (6)
where
a ~ A
>>">■ (7)
r and 6 are respectively the radius and density of the particle, and p ?
and y ? the gas density and thermal velocities of the molecules. Larger
particles with radii f^>-~ describe attenuating oscillations with respect to
the plane z = 0. Smaller particles sink asymptotically toward the plane
z = 0. Their z coordinate decreases e times within the time
< = J£f, (8)
which amounts to about 3 * 10 5 turns around the Sun for particles of radius
10" 4 cm.
Particle aggregation during collisions contributes considerably to the
speed of settling. Consider the settling of the larger particles, assuming
for simplicity that all others are immobile. Let the particle m absorb all
other particles it encounters on its way to the plane z = 0. Its mass
increment will be determined by the distance traveled:
dm = Anr^dr = — tcr'p dz. ( 9 )
From this we obtain the expression for the radius of the sinking particle:
dr = —
9 p dz
48
For a particle of variable mass m, equation (6) is replaced by
^( m .£) + M «r* +mA= 0. (11)
5979 26
or
f + a, ( , -?r£) i+, * =0, (12)
where coefficient a' is already dependent on %.
For small particles and z not small, the first term "i is very small ,
compared with the others" and can be disregarded. Furthermore, the second
term in brackets is small compared with unity. Therefore instead of (12)
we can write
T^r t h + «■ (*i - *)] =-(«,- **)•
(13)
Integrating, we obtain the time in which the particle sinks from z l toz:
l
/(«,. ,>«JLln(i-5L^.). (14)
The approximate expression (13) is not suitable for small z. Setting z ~~ z l /2 >
we find that, even for very small particles with r t ~~ 10~ 5 cm, the time for
the particles to settle to the central plane of the cloud when allowance is
made for their growth will amount to only about 10 3 revolutions of the cloud
around the Sun. In this time the particle radius will increase by
At the Earth's distance from the Sun o p mlO g/cm 2 and Ar -^ 1 cm.
Thus in the absence of turbulence, solid particles settle down to the
equatorial plane within a very short time, forming there a flat dust layer
of high density. When the density of this layer becomes critical gravitational
instability develops and numerous dust condensations are formed (see
Chapters 5 and 6).
9. Thickness of the dust layer in turbulent gas
The dust particles which formed as a result of the condensation of non-
volatile substances in the cloud originally traveled together with the gas.
Amounting to only 1% of the cloud mass, they had little influence on the
character of the random motion of the gas. As particle sizes increased and
random velocities in the gas decreased, the particles began to sink to the
central plane of the cloud. In the inner part of the cloud nearer to the Sun,
attenuation of random motions may have been less than complete thanks to
the perturbing effect of solar activity (corpuscular fluxes, magnetic pertur-
bations, etc.). For brevity we shall call such motions turbulent, not
investing the term with the rigorous meaning it has in the theory of turbu-
lence. These motions in the gas determined the relative velocities of the
solid particles and consequently the thickness of the dust layer. We will
first evaluate the relative velocities of the particles, making use of the
concepts of the semiempirical theory of turbulence.
* The time in which the particle velocity tends asymptotically to the value i, which is obtained from (12)
for£=0, is of the order of l/a\ i.e., about r • 10~ 3 turns. In the first phase of settling, the relative
variation of z in this time is small.
27
Let v be the mean turbulent velocity in the gas, t the mixing time, i. e.,
the time within which the turbulent eddy mixes with the surrounding medium
and p and v ff the density of the gas and the thermal velocity of the molecules,
respectively. The mean acceleration of the volume elements of the gas is
given by
fc~T- (16)
The gas carries solid particles along with itself. But its motion is not
transmitted to the particles completely. Let Ay be the rate at which particles
of mass m, radius r and density 6 lag behind the gas. Then the particle
acceleration can be taken as
f,~(» -£■)*■• (17)
The particle acquires this acceleration under the influence of the gas
pressure, and for a rarefied gas it can be written as
F An ?gV g r*bv p ff v
*>= m=— -£r — —zr Av > (is)
— &r3
Setting (17) and (18) equal, we obtain the relation between the particle
radius and its rate of lag with respect to the gas, Ay:
PgV At;
•0-t) '■
(19)
The particles' separation from the gas becomes significant for Ay ^ y/2,
i.e., for
|P >^=-V-- (20)
The relative particle velocities v p inside the cloud can be assumed to be
y-Ay. Then from (19) and (20)
Vp = v -to = v .-J*- m (21)
Urey concluded that dust is important in the attenuation of turbulence
(1958), He calculated the Reynolds number for a cloud a quarter of whose
mass consists of solid particles, taking the mean free path of particles
between mutual collisions as the characteristic dimension I and IdVJdR
(i. e., variation of the circular velocity along the path /) as characteristic
velocity. Urey obtained Re s» 70 for a Roche density* and the particle radius
* The concept of Roche density is directly related to the more familiar concept of the Roche boundary, which
characterizes the distance at which a fluid body of density p moving around a central body of density p and
radius R will disintegrate under the influence of its tidal forces: /? — 2.455/? vVq/P ■ Hence the expression
for the critical density P^ at which disintegration of the body occurs:
p A =14.8 Po (fl /^)3 = 14.8p* (
where
p* — 3 A# /4rJ7S « Po ( J ff /«)3 #
28
/•= 1 cm at the Earth's distance from the Sun. But no allowance was made
in this calculation for the fact that the dust grains are being carried off by
the gas and therefore do not reduce the mixing length I to the atomic mean
free path among the grains. If the particles are all the same size their
total resistance per cm 3 to the gas, according to (18)— (20), will be given by
Fn p = F !*-=!**!-=!£!--[*-. (22)
For a constant density p, of the solid matter, as the particles shrink their
resistance increases, tending to a limit which is only twice as large as the
resistance for r=r . Incidentally, the Reynolds number which Urey obtained
is proportional to r 8 . If we were to perform the substitution r=r (Urey takes
p = 10" 6 g/cm 3 and 6 =0.07 g/cm 3 for solid hydrogen, with r ^70 km), we
would obtain Re - 10 14 ,
The above relation (22) enables the attenuation of turbulent motions by
solid particles to be estimated. Particle acceleration by the gas g is
accompanied by a corresponding deceleration of the gas by the particles:
From (22) the characteristic time x p of damping of turbulence due to the
particles is given by
T ^-^75?=^17 T * (24)
Consequently attenuation of turbulence by solid particles becomes substantial
only when p p m p tf .
The thickness of the dust layer inside the gas is determined in the same way
as the uniform height of a heavier component of the gaseous mixture. In the
absence of macroscopic motions inside the gas it is uniquely determined by
the thermal velocities of the particles according to (5). But if the gas is
being vigorously mixed (e.g., due to convection), its components will not be
separate and will have the same uniform height. Hence the uniform thickness
H p of the dust layer in a turbulent gas lies between its minimum H , given
by the barometric formula (5) for the particle velocity v p according to (21),
and the uniform thickness H g of the gas, obtained from (5) for v—v :
H ~ = TT> *,=-sr- (25)
The upper limit of thickness of the dust layer can be substantially reduced.
The 'Stokes" velocity v s of particles sinking in the gas due to the accelera-
tion Z caused by the Sun's gravity can be found from (18) if we take g p =Z:
Uj = Z-t = A^ = A^, (26)
As long as v t remains greater than the turbulent velocity v in the gas, the
particle will drop continuously to the central plane. Therefore the half-
thickness of the homogeneous layer (H p /2) is less than the z determined
from (26) for v=v. In view of (5) and (21) we obtain
29
H <22 = 4-l2-=J-^±Lw = J_JLZ-ff . (27)
"p ^ w2 T r nave r P m nwx i> ff r 9
All particles with r>-^7~ r o must move toward the central plane, and the
larger they are the flatter the layer they will form there. The thickness of
the layer lies within the limits
H <H <— r ° +r i7 . (28)
In determining the scale of turbulence in a medium with differential
rotation from Heisenberg's theory, Chandrasekhar and Ter Haar (1950)
assume
Zoo J?, cccF,, t~~1/u). (29)
It seems, therefore, that one could set ut^ 1 in all the preceding relations.
Thus sufficiently large particles (r>r ) characterized by a relative indepen-
dence of motion will lie in a layer of thickness H p ^H pm .
In order for the critical density to be reached in the dust layer, the
latter must achieve a very high degree of quiescence and flattening. Accord-
ing to Ruskol (I960), after allowing for the gravitation of Sun and cloud the
density p in the cloud's central plane and its surface density o are related by
-V^3W. (30)
where tj = Po / P *; p* = 3 Af /4ir/? 3 . The value of 3(fj) is close to unity for a density
of the order of the Roche density; for p = 2p # , 3 = 0.9. _
For the dust layer it is necessary to set o = o^in (30). Taking 9$7> = i£/3,
we obtain
^-Ssjt- (31)
For gravitational instability in the terrestrial zone it is necessary that
~ 3 * 10" 7 (see Chapter 5). From (31) we find that for a p = 10 the particle
velocity v - 11 cm/sec. The layer's thickness H^a/ 9o must also be very
small: H/R&2 * 10" 6 . In the region of planetary giants conditions were more
favorable for gravitational instability; in the Jupiter zone one must have
v -270 cm/sec and H/R^ 10" 4 . The perturbations caused by solar activity
were more effective in the inner parts of the cloud and the associated
random velocities increased toward the Sun. Conditions were therefore
particularly unfavorable for gravitational instability of the dust layer within
the region of inner planets.
The size of bodies capable of separating out of the gas and forming a
flattened layer increases as random motions in the gas grow stronger. But
with increasing size the gravitational interaction of the bodies and the
relative velocities this interaction produces also increase. According to
(7.12), in a system of identical bodies of mass m and radius r, the relative
velocities are given by Yj?> For 6^3, bodies with r^2 * 10 4 cm have a
30
velocity of 11 cm/sec. On the other hand, from (21) we find that such bodies
will have the velocity v p = 1 1 cm/sec when y»380 cm/sec. Thus turbulent
velocities in the gas should be less than 380 cm/sec to achieve gravitational
instability in the solid body layer within the Earth zone. Otherwise, due to
the increased gravitational interaction of the growing bodies, flattening will
give way to swelling before the critical density of the layer is reached.
Table 3 gives the limiting values of the turbulent velocity v M in the gas at
various distances from the Sun. When the velocity of random motions in the
gas v>v M , gravitational instability could not have arisen in the dust layer.
TABLE 3
Mercury
Earth
Jupiter
Neptune
V g/cm 2 ....
1.5
10
20
0.3
9 tl'p
300
200
70
70
i? CP cm/sec . . .
0.4
11
270
50
r M . cm
7 -10 z
2 *10 4
6 ' 10 5
10 5
v M , cm/sec . . .
4
380
4 • 10 s
10 6
The values of i; cr in the third row were obtained from formula (31) and
represent the velocities of bodies at which the layer produced by them
becomes gravitationally unstable (p = 2.1p*). The values of r M in the next row
represent the radii of bodies whose velocities equal v cr due to their gravita-
tional interaction. The values of v M are turbulent velocities within the gas
obtained from (21) for o p — v CT and r = r M .* We see from the table that the v M
are very small for the planets of the Earth group and especially for the
Mercury zone, whose proximity to the Sun— a source of various perturba-
tions — makes it practically impossible to reach such small v M .
Thus it seems highly probable that gravitational instability of the dust
layer was present in the zone of planetary giants but not in the Mercury zone.
The influence of random motions of the gas on the solid material was
substantial only, so it seems, among the innermost planets (Mercury and
possibly Venus), within range of the perturbing effect of solar activity.
Where gravitational instability could not have arisen, growth of the bodies
must have been due to their aggregation in collisions.
For the motion of bodies in a gas the parameter e is several times greater than the value 6=3 adopted above
(see Table 11 in Chapter 7). Consequently, the values of r M and v x should be greater than given in Table 3
(about 2—3 times greater in the region of the Earth group and approximately 30% greater in the region of
planetary giants).
31
Chapter 4
TEMPERATURE OF THE DUST LAYER
10. Statement of the problem
One of the most important characteristics of the dust layer formed in the
equatorial plane of the protoplanetary cloud was its temperature, for on
this depended the chemical composition and mass of the layer. The chemical
composition of the dust layer largely determined the chemical composition
of the planets; the mass of the layer determined the size and mass of the
condensations formed inside it. Differences in temperature conditions
account for the division of the planets into two groups. It has been conjec-
tured (e.g., by Urey, et al.) that condensation of hydrogen could have
occurred in the large- planet region. It is natural to suppose that the
element most abundant in the cosmos should have been a major component
originally of the protoplanetary cloud. When studying the cloud's evolution
it is therefore particularly important to establish whether hydrogen could
have condensed inside it to the solid state.
Basing himself on the theory of common formation, Schatzman (1960)
considered the warming of the cloud by cosmic rays emanating from the Sun
at the stage of gravitational contraction, a period of intense electromagnetic
activity (for a solar radius twice as large as today). The turbulent magnetic
field enveloping the Sun prevented the rapid escape of cosmic rays from the
vicinity, and a large fraction of these rays was absorbed by the particles of
the protoplanetary cloud. For a total flux of cosmic rays of 10 33 erg/sec,
the temperature of the cloud was of the order of tens or hundreds of degrees
Kelvin. However, the parameters used are highly indeterminate.
Gurevich and Lebedinskii (1950) obtained the temperature distribution in
a uniform, optically thick two-dimensional layer extending in the direction R
and of constant thickness H along z, which is being warmed by ordinary solar
radiation for R = and is emitting in the z -direction. The temperature of
the layer decreases exponentially with R according to the law exp ( — i?/4ff),
and is very low at distances R many times greater than H f rom the source
of heat. The radiation, propagating inside the layer by diffusion, easily
escapes from the layer in the z -direction, and only a negligible fraction
penetrates to great distances R (Figure 1).
However, the dust layer revolving around the Sun in its gravitational
field was not plane-parallel or homogeneous. Near the Sun its thickness was
substantially less than at a distance, and its density decreased rapidly with z.
The Sun lay largely outside the layer, and its radiation, propagating almost
parallel to the layer, penetrated into the upper rarefied regions to great
distances, falling into the layer after scattering in these regions. Although
32
the scattered radiation did not amount to a great deal, it was sufficient to
prevent the temperature of the layer from dropping to extremely low values
(Safronov, 1962 b).
\ J / ^ - const
FIGURE 1. Warming of dust layer by solar radiation, according
to Gurevich and Lebedinskii (a), and warming of the layer due
to radiation by scattered particles lying in rarefied part of
layer (b).
For a numerical estimate of the warming due to solar radiation scattered
in the rarefied part of the layer, it is necessary to devise a reasonable model.
The absence of a clear idea of the genesis of the protoplanetary cloud makes
this task difficult. In Chapter 1 we noted that the different sections of the
cloud need not have evolved simultaneously. The inner parts of the layer
evolved much faster than the outer ones, but they could have been formed
later than the outer sections, as in Schatzman's theory. The role of solar
activity, which slowed down the flattening of the layer, and that of the
magnetic field, are also unclear. As these questions are undecided we will
consider the simplest model, a single optically thick dust layer having the
same coefficient of opacity x (per unit mass) at all distances from the Sun.
We assume the same intensity of solar radiation as today. Owing to the Sun's
heightened luminosity in the gravitational contraction phase, the temperature
of the dust layer in this phase must have been correspondingly higher. By
temperature of the layer we will understand the temperature of a black body
(black ball) placed at the given point, which is uniquely determined by the
mean intensity / of the integral (over all wavelengths) radiation at this point.
The temperature of real particles may be different. This applies in particu-
lar to particles at the surface of the layer which absorb shortwave solar
radiation in the visible region of the spectrum and emit in the far infrared.
However, in the central portion of the dust layer nearly all the radiation is
longwave as it has undergone repeated absorption and reemission by the
particles. Here it is possible to have local thermodynamic equilibrium in
which the temperature of the particles is almost identical with the black- body
temperature.
Essentially the problem breaks down into two parts: determining the
temperature distribution inside the layer for a specified value at its
boundary; and determining the boundary value. The first is relatively simple
33
to solve since the thickness of the layer is much smaller than the distance
from the Sun. As a result the temperature inside the layer is nearly
invariant with z (see Section 11). In practice, therefore, the problem
reduces to finding the temperature at the boundary of the layer. Its density
decreases indefinitely with z, and the concept of a boundary z x is arbitrary.
Whereas the position of the "surface" of the layer may be defined as the
smallest 2 for which t (z) & 0, the position of the "boundary" z x of the layer
must satisfy two requirements. First, in order for the equation used in
Section 11 for the stream of radiation to be valid up to z lt the mean free
path of the quanta must be considerably less than the half- thickness h of the
layer. This condition is met when the optical thickness t (2J, reckoned
inward from the surface, amounts to a few units. Second, outside the layer
and at its surface there is direct solar radiation and the black-body
temperature is higher than inside the layer. It decreases inward, rapidly
approaching its limiting value. This is the value which should be used for
T (zj). In practice this value is reached when t (z x ) is also of the order of a
few units. Quantity T (z) in Section 12 is obtained assuming gray absorption
in the layer, i.e., it is assumed that complete absorption (true absorption
plus scattering) is independent of wavelength. Then the mean free path will
be the same for all quanta, and when evaluating radiative transfer one can
therefore consider integral (over the entire spectrum) rather than mono-
chromatic radiation. For integral radiation, by contrast with monochroma-
tic radiation, radiative equilibrium exists since the radiation absorbed by a
particle is reemitted in the longwave region of the spectrum instead of
fading away. Consequently from the energy standpoint, in gray absorption
light propagation in the medium takes place in the same way as in pure
scattering. This makes it possible to use the results of the theory of diffuse
reflection and transmission of the light incident at the boundary of a plane-
parallel atmosphere. In the case of isotropic elastic scattering (no absorp-
tion), as one moves down through the atmosphere the mean radiation
intensity will tend to a definite limit which depends on the intensity of the
incident radiation and on the angle of its incidence. In the case of gray
absorption it is the mean intensity of integral radiation which must tend to
this limit.
11. Temperature distribution inside the dust layer
We will consider first a plane- parallel dust layer having a plane of
symmetry 2=0 and optical properties dependent only on the z- coordinate.
The integral radiation flux E inside the layer obeys the continuity equation,
which in the cylindrical coordinate system (R } z) has the form
dE,
dlT*-£+T*«+* = - (1)
Here Er and E M are the flux in the directions R and z, respectively. The
right-hand side is zero because there are no sources of energy in the layer
and the radiation, having undergone "true absorption," is again reemitted
in other frequencies. Consider the case of gray absorption and isotropic
reemission. The relation between the integral flux (over all wavelengths)
34
and the integral mean intensity of radiation J — ~\ldis can then be determined
directly from the diffusion equation
r 4* dJ
E — An dJ
* 3a dz «
(2)
where a— xp is the coefficient of absorption per unit volume. In the case of
particles of the same size, x will be independent of z and aocg. If, moreover,
the kinetic temperature does not vary withz, then pocr'''*'. However, in the
case of particles of varying sizes the smaller ones settle down more slowly
to the central plane; x will then increase with z (it is assumed that the
particle diameter is greater than the wavelength), and p will decrease more
slowly than e-**/* 1 . To simplify the calculations we can take
and h — const. Then from (1) and (2) we obtain, for z^O,
(3)
dm "*" R dR "*" dz* """ h dz ~ U '
The boundary conditions will be as follows:
/ = /,**-* for isi,,
dz
= for z = 0.
(4)
(5)
An approximate solution of equation (4) can be found for the layer with h<^R
(and correspondingly z^R). It is natural to expect that the value of /inside
the layer will not differ much from its value (5) at the boundary. Let us
write it as follows:
7 = A["i | M»> ■ "i(») . 1
(6)
Inserting this expression into (4) and equating the coefficients of different
powers of R to zero, we obtain equations for u. (z). Solving the latter and
choosing constants of integration such that the boundary conditions (5) are
fulfilled, we obtain
u x (z) = hp* [ Zl — z — h (e-*/* — e-i/*)];
u 2 (z) = h*p* (p + 2) 2 [(z x + h + &rV*) ( Zl - z) - j W - *") -
— tyzj + h + her'it*) (e-*/* — e~V*) _ A ( ze -'/» — v -',/*)]. ( 7 )
The substitution shows that the odd powers in square brackets in (6) will
drop out.
35
The quantity z x depends on x and roughly equals two to three half- thick-
nesses h. For zJR ~10" 2 we have u t (z) R* <10" 3 and u 2 (z) R~* ~10~ 8 . Thus in
the solution of (6) the second term plays an insignificant role while the third
term is negligible. The series converges very rapidly and the z-dependence
of / is determined practically only by the term with u x (z). The mean
radiation intensity / increases very slowly from the boundary of the layer
to the central plane. This increase is due to the fact that the radiation
reaching the central plane comes from a region at the boundary of
dimensions ~~ z x . Its intensity varies with R according to (5). The mean
value R~ p within such a radius around a point situated distance R f rom the
Sun will exceed the value R-*> by ~ z\jR\. Quantity /is related to the black-
body temperature T in the layer by the simple relation
T'=T> J ' ( 8 )
where & is the Stefan- Boltzmann constant. Thus we can assume that the
temperature of the dust layer is nearly the same throughout its thickness
and that it depends only on R.
In the following section we will adopt a more precise model of the layer
in which the density p is a function of R and h=pR . However, our conclusion
regarding the very weak dependence of / on z still holds. Indeed, if we take
a = a R~ n e-* /k , the constant factor (1 + n) will appear in the second term of
equation (4). The approximate solution of this new equation will be found in
the same way as above. It differs from (6) in the appearance of an additional
factor 1 — nip in u x (z), which even reduces u x (z) to some extent. If one further
takes h=$R, another additional factor (1 — zih) appears in the second term of
(4). The expression for u x (z) becomes more complex, but the order of
magnitude remains as before.
12. Temperature of the layer near the surface
The theory of diffuse reflection and transmission of light incident upon
the boundary of a plane-parallel atmosphere, developed by Ambartsumyan
(1942), Sobolev (1956)*, Chandrasekhar (1950) and others, enables us to
determine the density of radiation at large optical thicknesses as a function
of the intensity of the incident light and the angle of incidence. For isotropic
elastic scattering of uniform radiation incident at angle 6 to the inner normal,
the mean radiation intensity /, (x) will tend with increasing optical thickness x
(reckoned from the surface to the interior of the layer) to the following finite
limit:
/.(«>. ri=-3Gr*,G0*(p). (9)
where ji= cos 9; E^ (n) is the stream of energy of frequency v incident on the
surface per square centimeter per second in the direction ji, and <p (\i) is a
function given by the integral equation
fM = i+{ v . f M\lMl ; d v .<.
36
Tables of numerical values of this function computed by the method of
successive approximations are given by Sobolev and Chandrasekhar. For
isotropic scattering the function 9 (\i) is nearly linear: 9 (ji)^1+2ji. The mean
and maximum errors in this approximation are respectively about 2% and
less than 4%.
The relation (9) is valid for monochromatic radiation in any frequency v.
Since the black- body temperature of the layer is determined by the integral
radiation density in all wavelengths, it is sufficient to evaluate the mean
integral intensity / (00, \i) , without calculating /, (00, ^). Since (9) does not
contain the coefficient of absorption it is obviously valid for integral
radiation as well:
/(oo, rt = ^£(ri ? ( ( i) t (9 ( )
where
CO CO
Whereas relation (9) for /, (00, \i) holds only for pure scattering, relation (9')
for / (00, \i) is valid also for gray absorption. Indeed, if the total
absorption coefficient is independent of wavelength, the equation of transfer
will be the same for integral and monochromatic radiation. For integral
radiation radiative equilibrium will also obtain, since one is dealing with a
stationary case and the conversion of radiant energy into other forms of
energy is not assumed. Both scattering and reemission are assumed to be
isotropic.
In contrast with the semi- infinite atmosphere, a flat layer is symmetri-
cally illuminated on both sides. But provided the optical thickness of the
layer is large enough, for the same intensity of incident radiation the
radiation density in its central plane will be the same as in a semi-
infinite atmosphere at large 1, i. e., it will be given by (9 T ). In both cases
the radiation flux across any small area parallel to the layer will be zero,
as the amount of radiation reflected by the surface is equal to the amount
incident upon it. A layer irradiated from both sides receives twice the
amount of radiation, but its surface is also twice as large. That is, one
square centimeter of its surface reflects as much as one square centimeter
of the semi- infinite atmosphere.
The half- thickness h of the dust layer is small compared with the ,
distance R from the Sun. It is determined by the relative velocities of the
particles and depends on R . The simplest and at the same time most
realistic assumption is that k cc R . This corresponds to a relative particle
velocity proportional to the circular velocity (see (3.5)) and to a kinetic
temperature oc R' 1 ,
If the layer is very thin and h <^r R Q , one may disregard all effects
caused by the departure of the real cloud and incident radiation from the
ideal model for which relations (9) and (9') are valid. The calculation is
then particularly simple. Since the Sun's radius R Q < R , we have
ji = cos 6 < R Q IR < 1 an d Jj^jT) & 1 -f i? /i? ^ 1 .
37
Therefore according to (9'), to evaluate the temperature of the layer it is
sufficient to find the flux of solar radiation across one square centimeter
of the surface of the layer. The element of area ds = 2 y/R 2 e — C 2 dC of the solar
disk situated at height C will give the flux
«-£— a-uWiasp*.
where L is the Sun's luminosity, T § its effective temperature and a' the
Stefan- Boltzmann constant. The flux from the entire solar disk is given by
*=^a»T 2 J*^w=UV°' T '- (10)
o
From (8) and (9 1 ), for (p (p,) = 1 the temperature of the layer will be
However, if h is of the same order as R Q or larger, such a calculation
will not suffice. The radiation incident upon the layer, propagating nearly
parallel to its surface, travels a considerable distance within the outer,
rarefied portion of the layer. The fraction of radiation which reaches any
given point will depend to a large extent on the density distribution of the
matter on the way. If the layer is not plane- parallel and not uniform with
respect to R, the temperature must be calculated on the basis of a concrete
model. It should also be recalled that the Sun is not an infinitely distant
source. The optical thickness x (0) for 6 close to n/2 is smaller than
t (0)sec 9 = x (0)/n for an infinitely distant source, and it does not tend to
infinity for \i -► 0. Quantity / (00,(1) is greater in this case than given by (9 f ),
in which E (\i) = E'\i tends to zero for p-^O ( E' is the radiation flux outside
the layer across a perpendicular area). In the rarefied portion rays
propagate rigorously parallel to its surface (\i= 0); after scattering they
also penetrate into the layer, whereas (9') gives E (0) = in this case.
All these additional factors can be allowed for in determining E if one
computes the amount of direct solar radiation absorbed and scattered by the
particles in one square centimeter of the layer's surface— more precisely,
by the particles in a cylinder of unit cross- section with axis aligned with z.
Let us denote this quantity by E . For a plane- parallel atmosphere irradiated
by homogeneous parallel radiation, the incident radiation E is everywhere
the same and equals E . In the more complicated case we are considering,
the irradiation of the given area of surface is characterized in the first
approximation by the value of E . Since <p (^i)^l, inserting E in (9') we
obtain, instead of f E (?) dp — E , J(co)=j^ E and, from (8), the temperature
inside the layer. There remains a certain measure of inaccuracy due to the
inhomogeneity of the layer and of the radiation. In reality / (z t ) is
determined not only by the local value of E above this point but also by its
value in its vicinity, since the radiation is mixed as it penetrates deeper.
The mean value in the vicinity of r deviates from the value at the point by a
quantity of the order of r*/R % > in relative units. Qualitatively this deviation
is of the same nature as the increment in / (0) in the central plane z =-
38
over its vakfe / (z x ) at the boundary (Section 11). Although r exceeds h in the
rarefied region, on the whole the effect is slight, since h 2 IR 2 < 1.
Let us now estimate E . Obviously,
\ E =\adz i^r^ d *. (12 >
•©
where /'is the intensity of solar radiation at the point (i?, z) in the absence
of absorption, a-xp the absorption coefficient per unit volume and r the
optical thickness along the path from the elementary area ds on the Sun's
surface to the point (R, z). Since the integrand is very small outside the
interval (z lf z 2 ), and oo may be used conveniently as limits of integration.
From the considerations set forth in Section 11, we take
a = a e-'/\ h = $R. (13)
A light ray reaching the point ( R, z) from the point (0, C ) on the Sun's
surface situated at distance C from the layer's central plane will have the
following z coordinate at the distance R' from the Sun:
2 '^t + *l(2-Q. (14)
In view of the smallness of C and z compared with R, the distance between
the points (0, and (R, z) is practically equal to R. From (13) and (14),
the optical thickness along the path between these points i*= given by
R R
x (C,2) = j adR 1 = e-*/*e«* j a Q e-W R 'dR l = x (C,0) e~'IK (15)
o o
Let us substitute this expression for t(C, z) into integral (12) and first
integrate it with respect to z for constant C and R, taking t(C, z) as the
independent variable. Since from (15)
e~'f h dz = -
we obtain, dropping the prime of i?',
-qro)^' 2 )'
g-l CO
«^^.^,«-£* *-'* = ^ . (16)
-i "0
We further introduce variables u and v in place of i? and C:
in which case
«R 2 e kR 2 q ™ V u
39
and integral (12), in view of (16), becomes
*-4jv ^ dv • (17)
The intensity I' of the solar radiation at distance R outside the absorbing
layer can be expressed in terms of the effective solar temperature T , or in
terms of the black-body temperature T' at distance R from the Sun outside
the layer:
/' = oTj(^) 2 =4a'r*.
Introducing the value of /' into (17), from (8) and (9) we find the following
expression for the temperature under the surface of the dust layer:
r ,42vT
T* = T"t^afi\-gEZ*^ m (18)
■i«
For a R oc R-* we obtain
i
T* = T' K i^ B 1 "" ^ I ^' — y'yVy q 9 \
fcv
where k^R Q /$R. For n=- 1 this expression is easy to integrate and yields
the value
T =To (20)
where T is given by (11) and was obtained directly from the flux of solar
radiation across a small surface located in the plane z = on the assumption
that the radiation was not absorbed on the way.
From (19) it is seen that the temperature of the dust layer depends not
on the absolute value of the density but only on its gradient along R. The
faster the density decreases with R, the lower the temperature of the layer.
For w>l, T<T , while for n < 1, T > T . As the thickness of the dust
layer decreases the difference between the values of T for different n
decreases; at distances up to that of Jupiter from the Earth, for = 10~ 4
the difference amounts to less than two degrees when n varies from -1 to +1.
Since a oc p and R cc h, a R is proportional to the surface density of the
solid material in the dust layer. The latter may be regarded as roughly
constant up to the distance of Jupiter. Beyond the Jupiter zone, the density
begins to fall off sharply with R. Let us evaluate the temperature of the
dust layer for the following density distributions:
40
n = for H<i? ,
rc = 2 for R>R .
Then from (19) we obtain
- C r—
\~ V * / « P J e^Ej (V) + (V)" 2 [1 + *» — (t + M) e ( *-* o)B ] '
where Ei(i) is thfe function
_, dx
*.<*)= J *T
(21)
(22)
TABLE 4
P
Zone of
lO" 1
lO" 2
10~ 3
10-"
T
Mercury
Venus
186
130
107
85
43
23
11.6
7.3
136
91
75
58
28
14.8
7.4
5.0
119
76
61
45
20
10.5
5.9
4.2
115
72
57
42
16.9
9.5
5.4
3.9
115
71
Earth
56
Mars
41
Jupiter
16.4
Saturn
10.3
Uranus
6.2
Neptune
4.6
Table 4 gives the temperatures T obtained by numerical integration of
(19) and (22) for R equal to the distances of the planets from the Sun and for
different degrees of flattening p of the dust layer. In the low temperature
region stellar radiation becomes appreciable; its density is taken to
correspond to warming up to 3°K. The value p = 10 corresponds to the
density p»p*= 3 M /4nR z in the Jupiter zone, i. e., to a state of the layer
close to gravitational instability in this region. The last column gives the
values of the temperature T calculated from (11).
The table shows that for a constant surface density in the layer (up to
Jupiter), T -+ T when p -* 0. But if the layer is not very thin {k>R Q ) f its
temperature will be significantly higher than T . The radiation density in
this part of the layer decreases faster than R~* but more slowly than R~ 3 , in
accordance with (10). If one takes JqcR-', then when p increases from to
10" 1 the exponent p decreases from 3 to 2.2. The sharp falling- off of the
surface density in the region of large planets results in a faster decrease in
temperature with R. Here too, however, even for the smallest p the
temperature of the dust layer is much higher than that obtained by Gurevich
and Lebedinskii.
41
13. Warming of the layer by radiation scattered in the
gaseous component of the cloud
Above we examined the warming of the dust layer due to solar radiation
scattered by particles within this layer. But the dust layer, embedded
inside the gaseous cloud, is also warmed by radiation scattered in the gas.
The foregoing discussion still holds. The departure of molecular scattering
from isotropy does not substantially affect the numerical results, the
difference between the functions <p (p) for Rayleigh and isotropic elastic
scattering amounting to less than 3%. The thickness H of the gaseous
component is considerably greater than that of the dust layer. The quantity p
is larger than 10~ 2 for the gas, and the temperature of the layer, from
Table 4, is considerably higher. In deriving the fundamental relations,
however, the upper limit of optical thickness t (C, z) along R in (16) was taken
to be infinite. In a gas x (C, z) will be much smaller than in the dust layer,
and if it is small at the layer's boundary z x , then f e~ x di in (16) is less than
o
unity. Also, part of the solar radiation transmitted by the gas reaches the
dust layer. Thus for a dust layer surrounded by gas we obtain
E = E 0ff [1 - <r*. '.>] + E 0p e^ '.\ (23)
where E 0f and E 0p are the expressions for E in the form (17) for gas and
dust, respectively, and C^f?©/2. Since h p <^h g the second term is small and
cannot compensate for the dropping off of the first at small t (C, z x ) . From
(15) it is seen that for z x < h f one has t (C, zj&x (C, 0). The correction factor
to (17) is therefore roughly
S«l-e-^.<». (2 3')
In Rayleigh scattering the ratio of the amount of light scattered by a
single particle (atom or molecule) to the intensity of the incident light is
given by (Allen, 1955)
_128nE/ „«-l \« (24)
where N is a number in cm 3 and n the index of refraction. For molecular
hydrogen Born (Optics, 1937) gives the polarizability (n' — l)/4«tf= 8.2 - 10" 25 ,
The same value is obtained when one assumes, after Allen, that n = l. 0001384
for normal temperature and pressure and computes the corresponding value
of N=pikT. Thus
k =^=^ = 2.63 . 10" 5 X-*, (24' )
where k is now expressed in microns. From (15), for cr= const
x(C,0)=r^E,( p i?). (25)
42
For a gas the parameter [3 is determined by its temperature, and for T
independent of z it can be found from the barometric formula (3.5). Inserting
we obtain
^i*=^VW-
(26)
The density of the gas should decrease with z as e-*'/* 1 . But in view of the
fact that the gas temperature may have been higher at large z than at the
boundary of the dust layer, where it equals the temperature of the dust
particles, one can assume as before that the density decreased approximately
as a.e-^ h . Expression (26) with the value of T for the dust layer, and the data
in Table 4 give us two relations between p and T and make it possible to
determine both these quantities. They are given in Table 5.
TABLE 5
Zone of Planet
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
T, °K . . . .
145
100
84
67
35
18.4
9.3
6.1
P
0.019
0.021
0.023
0.024
0.033
0.033
0.033
0.033
*(V2*0. 0) •
0.85
1.04
1.15
1.3
1.5
«1
0.57
0.64
0.68
0.73
0.78
The values of t (#©/2; 0) in the third row of the table were calculated
from (2 5) for the surface gas density ct= 10 3 to the distance of Jupiter and
10 3 (R /R) 2 for greater distances, in accordance with (21). This value of a
corresponds to a total cloud mass of O.O46M , which is close to the value
0.05 M Q adopted by us after examining the rate of growth of planetary giants
(Chapter 12). The last row lists the values of the correction 1=^ for X = lju.
As three-fourths of the energy of solar radiation belongs to the region of
X < lju, the correction £''* for the cloud temperature is small. In the large-
planet region it is insignificant, the maximum value (in the Jupiter zone)
being -8%. It is slightly higher in the region of the Earth group. However,
x is computed only for Rayleigh scattering, without allowing for light
absorption by various molecules. In reality x should be larger and the
temperature correction smaller than given by |.
The data cited in Table 4 were obtained on the assumption that H oc R,
i. e., p — const. From Table 5 it is seen that this condition obtains in the
large-planet region where a falls off rapidly with R and T oc i? _1 . In the
Earth-group region T decreases more slowly, approximately as oci? -0 * 6 .
Therefore H oc R}- 2 and p increases with R. This departure from the
condition p= const ought to lead to temperatures higher than those indicated
in Table 4. The correction is small and opposite in sign to the correction
| v \ We will therefore limit ourselves to the uncorrected values of T given
in Table 5.
43
14. Condensation of volatile substances on particles
Lebedinskii has demonstrated that solid particles can warm up thanks
to the energy of random motion of massive protoplanetary bodies (1960).
The bodies acquire relative velocities due to gravitational interaction among
themselves. As they travel through the dust medium they undergo
deceleration and impart to the dust particles an amount of energy capable of
warming the latter by 5 — 30°K. Therefore hydrogen could not have
condensed on the particles in the region of the large planets. As Table 5
indicates, even in the early phase of the cloud's evolution, before the
formation of protoplanetary bodies, the temperature of the dust layer was
fairly high and as far away as Neptune hydrogen condensation on the
particles could not have taken place. Indeed, the condensation point of
gaseous hydrogen is related to its density (saturation vapor density) as
follows (Urey, 1958):
IgP = -^-Ig ** + 0.134 + 0.0363 7\ (27)
The actual gas density in the cloud's central plane is p — c^Aff , where H is
determined from (26).
At the distance of Neptune and Jupiter, for the foregoing values of a
hydrogen condensation is possible only at temperatures below 4°K and
slightly above 5°K, respectively. If we take o ff = 2400 in the Jupiter zone,
which corresponds to p =p* = 10~ 9 g/cm 3 , the condensation point of hydrogen
rises only to 5.5°K. For the value T= 35°K obtained above for the Jupiter
zone to drop to this level, the energy of solar radiation reaching this zone
would have to decrease 1600 times. It has been conjectured that the outer
parts of the cloud could have been screened due to thickening of the dust
layer in the inner region, for instance as a result of turbulence or convection
at the inner edge of the layer. However, in Chapter 2 we noted that the very
high temperature gradient along R necessary for convection cannot have
been achieved in this zone due to the Poynting- Robertson effect.
Certain perturbations could have appeared in the dust layer under the
influence of the strongest corpuscular fluxes ejected by the active regions
of the Sun. As yet it is not clear how efficiently these processes could have
transported solid particles to large values of z. It is not even excluded that
the fluxes flushed the dust particles out of the region.
Thus it seems that the mean radiant energy reaching the cloud was two
to three orders of magnitude greater than the energy at which hydrogen
could have frozen in the Jupiter zone. Therefore hydrogen could have
entered into the composition of the solid particles only in the form of such
compounds as CH 4 , H 2 0, and NH 3 . In the large- planet region all the latter
must have been in the solid state. It follows that the planets rich in free
hydrogen, Jupiter and Saturn, must have acquired it mainly in the closing
phase of growth when their mass had become large enough to hold the
acquired hydrogen.
44
Chapter 5
GRAVITATIONAL INSTABILITY
15. Fundamental difficulties in the theory of gravitational
instability in infinite systems
A medium is gravitationally unstable if newly developed density pertur-
bations in it. however small, increase indefinitely with time due to gravity
and disrupt the equilibrium.
Numerous works have been devoted of late to the problem of gravitational
instability. The interest stems not only from the great cosmogonic
significance of the problem, but also from the considerable mathematical
and fundamental difficulties encountered in connection with instability in
various systems. The linearized theory of instability, designed for a series
of concrete cases, reduces to Jeans' well-known criterion (1929), which is
in a certain sense evidence of its universality. On the other hand, it has
been stressed in a number of works that its derivation is faulty, as the
infinite homogeneous nonrotating medium considered by Jeans could not
have been in equilibrium. In nonequilibrium (expanding or contracting)
systems, small perturbations cannot lead to the formation of sufficiently
dense condensations, such as galaxies (Lifshits, 1946; Bonnor, 1957).
In stellar and especially in planetary cosmogony, long periods of time
present no difficulties. Here Newtonian analysis of bounded equilibrium
systems is expedient. The simplest problem will then be to study
instability in an infinite quiescent homogeneous medium. Jeans' criterion
can be treated as a first approximation that gives us, in the simplest cases,
the correct order of the critical wavelength of the perturbation responsible
for instability. Since among the forces counteracting instability allowance
is made only for gas pressure in the perturbing wave, Jeans' criterion gives
us the lower limit of the critical wavelength.
The main difficulty with Jeans' theory is due to a gravitational paradox:
for an infinite homogeneous medium there is no gravitational potential.
From Poisson's equation
^ + ^2+^5- 4 * G P [i)
forp^O, it follows that both the potential CD and the gravitational attraction
increase indefinitely with distance. This difficulty is circumvented in
Jeans' theory as well in its subsequent extensions by applying Poisson's
equation not to the entire medium but only to the perturbations, to the
departures of the density dp from its mean value p. It is assumed that in a
"truly" infinite, homogeneous, quiescent system, there should be no
45
gravitational attraction, as it lacks pressure gradient and accelerations.
Otherwise it would not be at rest.
Such an infinite system cannot be obtained by a limiting procedure from a
finite system (such as a spherical one) for 7?-*ao. Such a statement of the
problem cannot be applied to gravitationally bound finite systems, for which
it is necessary that Poisson's equation be satisfied in the Newtonian
approximation and its analog be satisfied in the relativistic approximation.
The simplest and clearest derivation of Jeans' criterion can be obtained
by considering the forces acting upon an element of the medium. Two forces
arise in the propagation of a perturbation wave: gravitational attraction,
related to the density perturbation 6p; and the gas pressure force, related to
the density gradient. For a plane wave at a point with displacement 5, the
former is given by (per unit mass)
F, = 4*G Po 6,
(2)
and the latter by
w,=-
i dp
p dx
P dx
c2 dip
<,+C
dx*'
(2*)
p = Po 4- s p» Sp =
<«
and the displacement I is assumed to be small. The velocity of sound is
denoted by c. For a sinusoidal perturbation
The instability condition
5 = ^ sin (urf + -^V
0X2 _ X2 *'
*F,>-*F,
(3)
(4)
leads to Jeans' well-known criterion for the critical wavelength of pertur-
bations:
X? =
"Gp *
(5)
Instability will develop for any perturbation of wavelength X > X c .
Further progress in the linearized theory of gravitational instability was
associated mainly with attempts to allow for rotation and the magnetic field.
Chandrasekhar (1955) considered the uniform rotation of an infinite
homogeneous system. Bel and Schatzman (1958) obtained a similar result
for a system of homogeneous density but in nonuniform rotation. They
analyzed perturbations propagating in a plane perpendicular to the axis of
rotation z, symmetric with reference to this axis and independent of z
(cylindrical). The instability condition they obtained has the form
. _ . 2o) d . D2 . , 4*2 f 2 „_
w
(6)
46
In these works as in many others dealing with rotating systems, Poisson's
equation is applied only to density perturbations. It is assumed that the
unperturbed medium is in equilibrium. But the question of how equilibrium
is established is generally disregarded. In contrast with quiescent,
infinite, homogeneous medium, in rotating systems a centrifugal force is
present.
One can suppose that this force is balanced by the attraction of the matter
contained in a cylinder of radius R which is infinite along the z axis. This
means that we are applying Poisson's equation to the homogeneous medium
along R and at the same time may not apply it along the z axis, for the same
reasons as in Jeans' theory— because there are no forces capable of counter-
acting gravity in this direction. The condition of equilibrium in the R
direction establishes the relation between p and w. For p = const, a> a =2nGp .
Inserting this value of co in (6), we find that the critical density necessary
for instability must be at least twice the actual density. Consequently, in
this case gravitational instability will not arise when perturbations propagate
in a plane perpendicular to the z-axis.
The instability condition (6) presupposes q = const, ©^ const. Such a
system cannot be in equilibrium. To achieve equilibrium additional masses
of nongaseous nature (stars), with a density p, dependent only on R, must be
introduced in the system (Simon, 1962 a). Even so, the instability condition
(6) will not be satisfied.
Thus for the systems under consideration the condition of equilibrium
(based, naturally, on the use of Poisson's equation) seems to be incompatible
with the condition of gravitational instability. A similar result was obtained
by Jeans for a finite spherical mass in equilibrium— it cannot break down
by gravitational instability into separate components. The theory of
gravitational instability in a rotating medium of infinite extension along z is
chiefly of mathematical interest, as there are no real systems to which it
could be applied. Nonetheless it represents an important step toward
understanding gravitational instability in real finite systems.
16. Gravitational instability in flat systems with
nonuniform rotation
Real astronomical systems of finite dimensions fall into two main
categories — spherical and flat. We saw that a spherical equilibrium
system of any finite radius cannot break down into separate clusters since
the critical wavelength is close to the diameter of the system. Instability
in expanding and contracting systems is examined in many works; we will
not be concerned with it, since the protoplanetary cloud belonged to the
category of flat rotating systems. In flat rotating systems in equilibrium,
by contrast with spherical ones, gravitational instability will arise if the
density of matter in the system exceeds the critical value. Due to the
complexity of the problem, however, for these systems no one has construc-
ted even a linearized theory of propagation of small perturbations.
Uniform rotation has been investigated by Fricke (1954), but he was
unable to avoid arbitrary assumptions. We have demonstrated that Bel and
Schatzman's attempt to apply condition (6) to flat systems is untenable: too
low a value is obtained for the critical density. It was found that it is
47
possible to effect a transition from two-dimensional cylindrical rotating
systems to flat ones by multiplying the term 4nGp in the left-hand side of (6)
by the function f (k/H) < 1; the latter function was calculated (Safronov, 1960 c).
It will be seen that condition (6), like the instability condition (4),
represents the balance of the forces acting upon an element that has been
shifted radially by the perturbing wave through a distance 6R=1 without
change of angular momentum with respect to the center of the system. Since
the maintenance of equilibrium in a homogeneous medium with nonuniform
rotation requires us to assume the presence in the system of additional
masses of a different nature (such as stars), it is simpler to consider the
variation of F g and F c at a certain fixed point in space (at the given R) than
to track the displaced element.
From Poisson's equation it is easy to find the attraction per unit mass of
an infinite cylinder:
-2GmjR,
(7)
m being the mass enclosed in a cylindrical layer of radius R and height 1 cm.
For radial perturbation the mass m will change by 6m=— 2nRp6R. Consequently
8f, = — 2G§mjR = AnGfiR. ( 8 )
This expression is the left-hand side of (6). The centrifugal force is given
by F e =(a 2 R = k 2 /R 3 , where k=a>R 2 is the angular momentum density. The
element moves over the distance R with the angular momentum it possessed
in unperturbed displacement at the 1 distance R—6R. Therefore
v.— i^Sr— lrar(«^»- (9)
Since in real systems ^ R >0, the force 8F e is negative and tends to
return the displaced element to its previous position. As we know, rotation
stabilizes the system. Expression (9) represents the first term in the right-
hand side of condition (6) for bR = 1. The second term of (6) is the gas
pressure gradient due to perturbation, and it too represents a force acting
opposite to the displacement. The last term of (6) can be disregarded,
since only perturbations with X <i?, for which this term is much smaller
than the preceding one, are of practical interest.
When one passes from a system infinite along the z~ direction to flat
systems, only the term related to gravity in the left side of (6) changes.
It is no longer worthwhile to use Poisson's equation to determine the
component of gravitational attraction along J? of a ring of density 6p, as it
now includes the additional term d 2 6y/dz 2 . It would therefore be more
expedient to calculate 6F ff directly. The expression for 6F 9 is somewhat
cumbersome, as it contains elliptic integrals which, moreover, lie under
the integral sign (Safronov, 1960 c). However, in the case of a ring the
first- order term which interests us in &F g is equal, for X <g :R , to the value
of 6F g for an infinite cylinder perpendicular to R and z and having a cross
section equal to that of the ring containing the point R Q . We will confine
ourselves to the simpler evaluation of 6F for the cylinder, which corresponds
to the case of a plane wavefront. Consider a sinusoidal wave perturbation
having amplitude 6R at the point R :
48
S = Si? cos -y- , Sp = — p ^ = -j— sin — p, (10)
where r=R~R . From (7), the attraction exerted at the point R by an infinite
elementary cylinder of cross section drdz and density dp whose generator is
perpendicular to the R and z axes is given by 2Gtydrdzi'\/r 2 -\- z 2 ; its component
along R is given by
^ = r2 + Z 2 —T— Sm — 72+12- (11)
For p we take its value averaged over z within the homogeneous thickness 2h.
The limits of integration over r will be +X/4, the maximum distance reached
by the perturbation, which has a first maximum \ at R , and correspondingly
- V/4.* Then
+X/4 +A i/4
8 ^ = 4nGp»fl ^ J sin?f^=16 n Gp8flj S in?farctgidr. (12)
-X/4 -A
Setting
we find that
where
'=*> 5=F=<. (13)
BF, = 4rcGp/(C)5fl, (14)
/W= J sincere ctg^ da:. ^^
Thus the gravitational instability condition for a flat system in nonuniform
rotation can be written as
4«Gp/(C)>!(u.fl7+^ 1 . (16)
The attraction of the central body (Sun) is present in the form of the
function <${R). The function /(C) has the following values:
c. . .
. . 0.2
2
4
6
8
10
14
20
/(C) .
. .0.96
0.64
0.43
0.34
0.28
0.23
0.172
0.124
Hence the correction to the critical density is significant and depends on
the ratio of the wavelength of the perturbation to the thickness of the layer.
Let us find the value of C for which the critical density necessary for
gravitational instability is minimum. From (3.30)
''' The lower limit of integration rjwill depend on the nature of the perturbation (single wave or train), and is
not completely defined. But the result is not strongly affected by this: for r±=— 3X/4 the value of lF g will be
10°fo higher than obtained above, and forr^ = — conot more than lS^o higher. It is interesting to note that in
flat systems, unlike systems infinite along z, the perturbation Bp begins to excite the gravitational force %F g at
the point B a not at the instant when it reaches R but rather when the perturbation appears at any distance,
however large, from i? . But the maximum perturbation occurs for maximum displacement 8/? to R .
49
H=± = V^*L.3, (17)
where
i
2 J iA ^7"' ~* ' p "*»• ( >
Evaluation of the integral gives us the following dependence of 3 on q/q*:
p/p* .... 1/3 4/3 10/3 5 10
3 0.66 0.87 0.94 0.96 0.975
Substituting ff = tyC and 9*77fi = c 2 /T in (17), we obtain
l^!£!-4 K Gl^o (19)
where
For a system whose rotation is determined mainly by the attraction of
the central mass (solar system, outer parts of the Milky Way),
o>R 2 =\/GMR, ?£ (o>fl 2 )' =fl,»=:l«(;p*. (20)
Then the stability condition (12) can be written as
P>/-(0(£+g£). (21)
The quantity q represents the density of a homogeneous layer of
thickness 2h. A real rotating cloud with exponential density distribution
q (z) will have a lower concentration toward the 2=0 plane. It will produce
the same bF g along R as a homogeneous layer would for q > Q .
Calculations show that Q&0.9 q and is only weakly dependent on C.
Recalling this and using the numerical values listed above for /(C) and 3,
one can find the critical value q satisfying the instability condition (21).
The results of calculations for y= 1 are cited below and in Figure 2.
C 4 6 8 10 15
p 0cr /p* ... 6.8 2.3 2.1 2.2 2.4
Thus the critical density required for gravitational instability, which
depends, as we know, on X, is minimum when the wavelength of the pertur-
bation is eight times the cloud thickness H. As X decreases the critical
density increases due to the increase of the second term in (21), which is
related to the usual Jeans criterion. As X increases the main factor in (21)
becomes the first term, which is related to the rotation of the system.
50
Here the critical density increases due to the increase of the function f~ l (Q ,
which shows how many times smaller the attraction of a flat ring is than the
attraction of a tube constructed on this
ring which is infinite along the z- direction.
The minimum critical density Q cr = 2.1 q*
is more than 6 times greater than the
critical density o*/3 obtained by Bel and
Schatzman for the two-dimensional case.
It is also larger than the value obtained
by Chandrasekhar for uniform rotation.
Consider the influence of the magnetic
field and viscosity on the instability con-
dition. If the perturbation is propagating
perpendicularly to the magnetic field
(in our case this corresponds to a toroi-
dal field), then instead of c 2 in the insta-
bility condition (16) we have the sum
A///
FIGURE 2. Dependence of the critical density
for gravitational instability in* a flat layer ro-
tating around a central mass M on the wave-
length of radial (ring) perturbation:
c 2J rvl, where v a = #/v/47ip is the Alfven
H— layer thickness; p" = 753*
velocity.
In the linearized theory the introduc-
tion of a nonzero viscosity for the
medium leads to the exclusion of the term related to the system's rotation
from (16), while the introduction of magnetic viscosity leads to the exclusion
of the factor v* a in the last term (Pacholcayk and Stodolkiewich, 1960). Allow-
ance for the thermal conductivity causes the velocity of sound to change from
adiabatic to isothermal (Kato and Kumar, 1960). The coefficients of viscosity
(ordinary and magnetic) and of thermal conductivity do not enter into the
instability condition. However small they are (provided they are nonzero),
the corresponding terms will not appear. Since under real conditions the
viscosity, although very low, is not rigorously zero and the electrical conduc-
tivity is not infinite, it is sometimes formally inferred that neither the
system's rotation nor the magnetic field affect the stability of the medium,
and that the ordinary Jeans criterion (or Ledoux criterion for a flat system)
is to be used. This result is strictly due to the fact that the perturbations
are assumed to be infinitesimal. They develop over an indefinite period, and
thus the viscosity of the medium and the thermal conductivity would eventua-
ally produce the indicated effect. Here, however, we encounter a difficulty
which is not accorded the attention it deserves. In general, a viscous me-
dium with differential rotation is not in equilibrium. Hence to state the
problem of the instability of such a medium with respect to infinitesimal
perturbations assuming that the unperturbed medium would be in equilib-
rium is wrong in itself. Moreover under real conditions perturbations are
always finite, and in many astronomical systems the viscosity is usually so
low that it is not a factor. The instability criterion for such systems should
have the form (16) with the addition of v\> i.e., both rotation and the magnetic
field are important in this relation.
A similar situation obtains in regard to the influence of the magnetic field
for an infinite electric conductivity, when the field is not rigorously perpen-
dicular to the perturbation. Formally the quantity v a drops out of the instability
condition even for a very small component H x of the field along the direction
of propagation of the wave. Yet in reality for finite perturbations a field
nearly perpendicular to the perturbation will set up a resistance to it,
contributing to the stability of the medium.
51
Thus for finite perturbations the gravitational instability conditions
differ in a number of essential respects from the criteria obtained assuming
infinitesimal perturbations. This fact must be taken into account in cosmo-
gonic applications of the theory of gravitational instability.
17. Growth of perturbations with time
It is usually understood that for X > X c the perturbation will lead to
unlimited compression of the flat layer. In particular, this conclusion is
drawn by Simon (1962b). Looking at the equation of motion, Simon found
that for a sufficiently large time t the density increase at the center of the
layer is approximated by the function t~^e kt , However, this conclusion is
wrong. The equation studied by Simon applies only to small perturbations
and is unsuitable for large periods of time in which the density becomes
significantly higher than its initial value. For displacements of the same
order as X, expression (2) for the gas pressure at a point with initial
coordinate x and displacement \ (z) has the form (Safronov, 1964 a)
8 P J dp dp _ / p \t-i {dpld?) dp / 99 \
?dpd(x + l)— \pj f t dt\dx' K }
P
where Y~c p jc t .
The continuity equation gives the relation
P (*+£) = *• (23)
Introducing p into (5) as in the above expression and setting (dpidp) Q = c 2 , we
obtain
The equation of motion will therefore be as follows:
s-^+^+srs- (25)
Simon's equation lacks the term dljdx, or, which amounts to the same, the
factor (p/p ) 1+T for d 2 ljdx 2 . As long as the displacements are small, p»p ;
dH g
for X>X tf the right-hand side of (25) is positive and
<*/2*)«
increases with £. The perturbation is constantly increasing in strength.
But when p becomes distinctly larger than p , the second (negative) term in
the right-hand side increases more rapidly than the first and contraction
ceases. In the case of sinusoidal perturbation and ? = 1, by the time p = p X/X c
the acceleration d 2 ljdt 2 = and the rate of contraction begins to fall off.
Simon's inference that instability develops even for X<X a is based on a
misunderstanding.
A flat layer cannot contract indefinitely, even if all the heat generated is
emitted in the process (isothermal contraction). According to Ledoux (1951)
an infinite flat isothermal layer of specified surface density a has the
following thickness when in equilibrium:
*=£&• (26)
52
In reality H does not depend on 7 ( c z oc y in the numerator) and is determined
by the temperature of the layer. The above expression refers to an isolated
layer. But a layer formed by gravitational instability will be surrounded by
an infinite attracting medium which stretches the layer, increasing H and
decreasing the density Q e in its central plane. Recalling that
o = Pc ff = p X,
we find from (26), in view of (4), that
^<^ 2 - (27)
For X > X e one can take the equality sign above, since H <^ X and the attraction
of the surrounding medium is small.
An isothermal flat layer in equilibrium is not bounded. Consequently,
for a perturbed region with initial dimensions X the contraction stage should
give way to a stage in which its outer portions expand and fuse with the
surrounding medium. According to the linearized theory of instability, for
X > X c the rate of wave propagation becomes imaginary, and no perturbation
will propagate beyond the area of developing instability. However, the
picture changes radically when the process deviates from linearity. No
closing up of the localization of the perturbed region takes place in the case
of a flat layer. The expansion wave penetrates to the surrounding medium,
where it produces a perturbation which continues to travel on.
The foregoing considerations do not rob Jeans' theory of gravitational
instability of all cosmogonic significance, but they do significantly alter our
conception of the nature of the development of instability. A single wave
perturbation is not sufficient for the density to increase indefinitely. It
merely leads to the formation of a flat layer. But if a new perturbation of
wavelength X' > X' c were now to travel along this layer, it would again give
rise to gravitational instability, leading to the formation of a contracting
cylindrical region (quantities referring to the flat layer will be designated
by a prime). According to Ledoux for y = 1
*=£• (28)
Earlier we found that when a perturbation travels along the layer the
gravitational attraction 5F^ is given by expression (14):
tf; = 4*Gp./(C)S. (29)
The development of perturbations inside the layer proceeds as in a medium
infinite in all directions, except that when describing them the gravitational
constant G should be replaced by G/(C). The presence of the extra factor of
two in the numerator in Ledoux's criterion (28), compared with Jeans'
criterion (5), is due to the fact that for y = 1, from (26) and (28), X'JH — n
while /(n) = 0.5.
If the thickness H remained constant throughout the process of develop-
ment of the perturbation with X' > X' e inside the layer, then, as in the
preceding instance, the perturbation could not have caused unlimited
53
contraction. For maximum density growth we would have an expression
similar to (27) with an extra factor /(#'/#;// (X'/#) on the right, of the order of
two. The width of the contracting zone would have reached the value H' < H.
In reality contraction proceeds in both directions and the configuration tends
to an infinite circular cylinder.
The cylinder represents an intermediate case between a flat layer and a
sphere. Earlier it was shown that for any y a flat layer is stable. On the
other hand it is well known that for 7 < 4/3 a sphere will be unstable and will
contract indefinitely. Applying similar arguments to the cylinder, one finds
that the critical value of the adiabatic index y= 1. Indeed, for a cylinder of
radius r
F — 2Gm {r)
P p dr ~~ p r r '
If the cylinder is subjected to radial contraction, then m(r) «• const, oc r" 1 , and
F s oz r~\ ^ocr"^)-i, (30)
For instability it is necessary that F g change faster than F , i. e., that
2( T _1) + 1<1 and T <1. (31)
For all v >1 the cylinder will be stable under radial contraction. The case
7=1 corresponds to neutral equilibrium: if there was equilibrium prior to
contraction, contraction will not disrupt it. It is therefore desirable to
determine what equilibrium configurations obtain for an isothermal cylinder.
For an infinite cylinder the condition of equilibrium in the radial direction
r
dp = —g ? dr, where g=?^ll f m (,-) = 2* j r 9 dr (32)
o
leads to a second-order differential equation (Safronov, 1964a; Ozernoi,
1964):
rf2p I dp I /rfp\2 . 2wGp2 n
dTS + T Tr~j{Tr) +-^- = a (33)
This equation can be reduced to the Euler equation. For the boundary
conditions
r = 0, P = p(0), g = 0. (34)
Its solution is given by
P-p(0)(l-faV 2 )- 2 , (35)
where
*'=^. (36)
54
The mass of a unit cross section within radius r is given by
The total mass of a unit cross section of the equilibrium cylinder,
m = m e = 2c s jG (38)
is independent of q(0) and has a unique value.
Hence the isothermal cylinder is not an equilibrium configuration: it
contracts for m > m c and expands for m < m e . Equilibrium is possible only
for m exactly equal to m c and, as we saw earlier and as is also evident from
(35), it is neutral with respect to radial contraction. In the latter solution
q(0) can be varied at will. Correspondingly a varies but m=m c continues to
hold. Expression (38) for m c can be obtained more simply with the aid of the
virial theorem up to a factor of the order of unity (McCrea, 1957).
Let us return to the flat layer. For the critical wavelength the mass of a
unit cross section of a contracting band, from (26) and (28), is given by
m = V c9e H = Xf 9 "=^ = w ,. (39)
Consequently the gravitational instability condition in a flat layer (X'>r)
leads to m>m e , i.e., it is also the condition for unlimited contraction
stemming from the instability of the cylinder. We note that the condition for
instability for a cylinder with respect to perturbations along its axis,
obtained by Dibai (1957), is also fulfilled:
2 2
~ 4G *
(40)
where \i l = 2.4. The cylinder breaks up into separate clusters which contract
even more rapidly.
A similar picture will obtain in the case of an annular perturbation inside
a flat rotating layer. The supplementary term in (16) related to rotation
will be a significant factor only at the early phase of development of the
perturbation when it is of the same order as the next term characterizing
the gas pressure. But when the width 2r e of the ring shrinks by more than
half, the increase in F c (in absolute magnitude) will be distinctly slower than
the increase in F pt as F c oz £<X/4 while F p oc rj 1 (from (30)). Since condition
(16) is fulfilled at the start, it should certainly be fulfilled later, ensuring
the further growth of the perturbation.
At the initial phase, when the perturbation is still small, it increases
exponentially as £ = $ e (O ',
" 2 = 4«G(p ln - Pcr ) / <q - 3b» (p. n /p* - Pcr /p*) / (C), (41)
where a> e is the angular velocity of revolution around the Sun. The pertur-
bation will develop very rapidly even if pin is only slightly greater than p cr .
Let p cr =2.1 P *, p in =2.2 P *, and /(C) = 0.28. Then w=0.29to c and at the Earth's
distance from the Sun the time required for e-fold growth of the perturbation
55
is given by g)' 1 ^ 0.55P, where P = 2n/a) c is the period of revolution around the
Sun. Within 10 revolutions the perturbation £ will increase 10 8 times.
Twenty years would be sufficient for even a very smallperturbation ( S = 1 cm),
set up by a corpuscular stream possessing energy characteristic of large
solar flares, to grow to a considerable size (S^/felO 8 cm) at the Earth's
distance from the Sun.
Thus cosmogonically the gravitational instability in the flat layer revolving
around the Sun developed within a very short time, of the order of a few
tens of periods of revolution.
56
Chapter 6
FORMATION AND EVOLUTION OF PROTOPLANETARY
DUST CONDENSATIONS
18. Mass and size of condensations formed in the dust layer
The inference that the dust layer revolving around the Sun disintegrated
into a large number of dust condensations was first stated, independently
and almost simultaneously, by Edgeworth (1949) and by Gurevich and
Lebedinskii (1950). Basing himself on Maxwell's well-known research into
the stability of Saturn's rings (1890), Edgeworth conjectured that for a
density of 0.04Q*the layer becomes unstable, with random eddies developing
inside it; these eddies give rise to density fluctuations which grow and
transform for q > 3q* into roughly spherical nondisintegrating condensations.
However, the probability for such large random fluctuations is extremely
low. We saw earlier that gravitational instability develops inside the layer
at significantly higher densities: Q > 2Aq*. Maxwell's result is rigorously
valid only for material points situated on a ring and is not applicable to a
large number of particles colliding with each other and forming a virtually
dense medium. The Maxwellian ring breaks down when the amplitude of
oscillation of the material points equals the mean distance between adjacent
points and collision between them becomes possible. But in the layer,
collisions between real particles have no effect whatsoever on its stability.
Gurevich and Lebedinskii estimated the densities and sizes of condensa-
tions after analyzing the energy aspects. They determined the size and
density that a spheroidal region formed in an unperturbed disk must have if
it is to hold together by internal gravitational attraction when the material
of the disk surrounding it is removed. The fundamental condition was
obtained from the virial theorem, both the random relative velocities of
the particles and the ordered velocities associated with the cloud's differen-
tial revolution around the Sun being taken into account. It was found that the
density of this spheroidal region (condensation) must be one order greater
than the "spread out" density q* of the Sun, and that sizes in the plane of the
layer should be 1 3 times greater than sizes at right angles to it, i. e., than
the uniform thickness H of the layer. The first result agrees with estimates
of the Roche density p R . The second was not known previously.
This method gives the following values for the mass m and semimajor
axis a of the condensation:
"o*^. ao^^i. (l)
where a is the surface density of the dust matter in the layer.
57
A certain inaccuracy is introduced when the condensation is assumed to be
spheroidal. The actual condensation could not have formed from a spheroidal
region, since different directions inside the central plane of the layer were
not equivalent. As a density estimate applies to an isolated condensation, in
the presence of surrounding medium it will give an excessively high value
for the density of the condensation compared with that of the background. It
is evident that such large density fluctuations could not have arisen in the
absence of gravitational instability. Of all the possible perturbations it is
best to consider radial annular perturbations, as they are the only ones
which do not disintegrate upon differential rotation and which can increase
in intensity during many periods of revolution around the Sun.
In Chapter 5, Section 16, we saw that for a density q > g cr radial pertur-
bation will lead to the formation of a contracting ring. In the contraction of
the ring the angular momentum of its material with reference to the Sun is
conserved, and therefore its orbital velocity is changed. The outer half
moves toward the Sun and accelerates, while the inner half moves away and
slows down. When the width of the ring shrinks by one half, the linear
velocity of rotation of all its parts becomes uniform, and by the time it
shrinks to one quarter the ring revolves as a rigid body. When the width of
the ring is much smaller than its radius, the condition that it disintegrates
into separate condensations is close to the condition of disintegration of an
infinite cylinder. The condition for gravitational instability of an infinite
homogeneous fluid cylinder of radius R with respect to longitudinal pertur-
bations was obtained by Dibai (1957) in the form
u hi
(2)
Taking the mass of a unit cross section of the cylinder to be equal to the
mass of unit cross section of a ring with initial width C#,
and, from (5.17),
we obtain
ic P fl; = Coff
X _2* „ /SC3* ^V. /oX
For C = 8, nj = 2.4, r = 5 A and 3 = 0.92, we obtain X = 3/?
The minimum mass of the condensation is given by
(4)
For /?„ we can take the geometric average between the half-width ^Hj2n x of the
ring and its half- thickness (~~///2) at the instant of disintegration. Quantity n^
shows how many times the width of the ring has decreased at the instant of
disintegration. Then
5979 58
^^•rf-^w; m °- (4')
For Po = 2.5p*, C = 8 and n } - 4 the minimum mass of a condensation turns out
to be three times greater than the mass obtained according to the method of
Gurevich and Lebedinskii. The initial radius of the condensation in the radial
direction is given by
2n, n,p ft o
For the same values of p and n t we obtain a= 0.8a . The minimum size of a
condensation in the direction of orbital motion will be 1.5 times greater than
radially. The actual size may be slightly larger. Attention should be drawn
to the close agreement between estimates for mass and radius as obtained
by the different methods. Note, too, that gravitational instability inside the
dust layer and dissolution of the condensations could have taken place for
different values of the parameters | f n x and p in (4'). Thus there could have
been considerable divergences in the initial masses of the condensations.
Henceforth we will use the following mean initial values:
m o = X2* a o = 2F' (6)
For the terrestrial zone this yields m Q = 5 • 10 16 g and a = 4 ■ 10 7 cm for
a= 10 g/cm 2 ; at the distance of Jupiter the values are respectively 10 22 g
and 10 ro cm for ct= 20 g/cm 2 .
The internal gravitational forces of the evolving condensation exceed the
external forces. It therefore begins to contract until its gravity is balanced
by internal pressure and by the centrifugal force, which increases with
contraction. The decrease in the relative velocities of the particles caused
by inelastic collisions is accompanied by contraction of the condensation
along the axis of rotation. However, frequent external perturbations
resulting from encounters and collisions among condensations tend to prevent
unlimited slowing down of the particles and unlimited flattening of the
condensation.
The equatorial radius r of the condensation and its angular velocity of
rotation w before contraction (subscript 0) and after contraction (subscript 1)
are related by the condition of angular momentum conservation and the
equilibrium condition:
rsu>,» = rfu> 1 ,
rW=&m, (7)
where m is the mass of the condensation and £ is a coefficient which depends
on the form of the condensation. Hence
" ZGm*
(8)
For a homogeneous spheroidal condensation revolving as a rigid body
(Maclaurin spheroid), te 1/2 for the axial ratio c/a = 0.6, tel for c/a= 1/3,
and te 2 for c/a= 0.1. In the other extreme case of a Roche model (nearly
all the mass concentrated at the center), 1= 1. Therefore without introdu-
cing a major error one can assume that g» 1.
59
Thus the size of condensations after contraction is determined by their
rotation at the initial instant. To calculate the rotation of condensations we
can turn to the premises regarding their genesis which we outlined above
while evaluating their masses. According to the first premise, the angular
momentum of the region from which the condensation evolved (and every
volume element of which is in unperturbed circular Kepler motion around
the Sun) with reference to the center of the condensation can be taken to be
the mean rotational momentum of the condensation. If that region is a flat
uniform circle, then its mean angular velocity of rotation around the center
will be
% = ±rolV=±±(Vfl)=± u>c , (9)
where V and (o tf are the linear and angular velocities of revolution around the
Sun. If instead of a circle one takes a uniform spherical region, the coeffi-
cient will be 1/5 instead of 1/4 (Artem'ev, 1963). The mean rotation of the
region is direct, i. e., in the direction of revolution around the Sun. The
direct rotation of this region had already been inferred by Prey (1920) and,
based on a more accurate analysis, by Rein (1934). To obtain the angular
momentum of a noncircular flat condensation, it is necessary to integrate
the relationship
*i+'($d-.(*-W) < io >
over this entire region; the above relationship describes the angular
momentum density with reference to the center of the condensation of an
element situated at the point (x, y) and moving along a circular orbit around the
Sun with velocity V (R) (Edgeworth, 1949). The ( a, b) frame of reference is
nonrotating. Its origin lies at the center of the condensation. At the instant under
consideration, the x and y axes lie along the orbit (in the direction of revolution)
and along the radius vector, respectively. For an ellipse with semiaxes ( x, y )
along x and y , the mean angular velocity is given by
2a2— 62 (11)
a) o = 2(a2 +6 2) u) * = ga V
The condition that the ratios of the kinetic to the potential energy at the ends
of the axes be equal leads to b/a = 3/4, while the condition that the total
particle energies be equal leads to b/a= 1/2. This gives a equal to 0.5 and
0.7, respectively.
If the region that is contracting inside the Sun's gravitational field is
small compared with the distance R from the Sun and symmetric with
reference to the x or y axis, then its angular momentum will be conserved
(Hoyle, 1946). This allows us to use expression (8) to evaluate the radius
of the condensation (1) after initial contraction. Assuming that ct^l/2 and
1= 1 and taking m and r according to (l), we obtain
« 2 rl "J 3 2
60
The rotation of a condensation formed in the disintegration of a ring is
particularly easy to estimate for n^ 4. In this case the condensation will
rotate as a rigid body with angular velocity m c of revolution. Therefore in
expression (12) we now have a^l. From (4 ! ) the condensation mass mis
roughly three times greater than m , taken in (12) in accordance with (1),
Therefore r,/r «l/3 and not 1/4 as in (12).
Thus the initial contraction of the condensation causes its initial radius
to shrink by a factor of three or four and its density to increase by one
order or more.
19. Evolution of dust condensations
In the next phase the evolution of the condensations was considerably
slower. In Edgeworth's view, it was determined by the tidal forces of the
Sun which gradually slowed the condensation's rotation, thereby enabling it
to contract along r. Due to the smallness of the condensations, however,
the time span of this type of evolution would have been very considerable.
The condensations must have contracted far more rapidly by collision and
fusion. Thus when two condensations that have collided centrally combine,
their mass doubles, while the angular momentum density remains as before.
From (8), the radius of such an aggregate should shrink by a half and its
density increase 16 times. For such rapid evolution the influence of the
Sun's tidal forces on the rotation of the condensations would be negligible.
According to Gurevich and Lebedinskii, initially the condensations
traveled along nearly circular orbits. Since only those condensations that
lay along nearly the same orbit could have combined, aggregation (in the
authors' view) kept the angular momentum density constant. The aggregation
process lasted over the period Pie (i. e., of the order of 10 5 years in the
Jupiter zone) and led to the formation of "secondary condensations" with
masses of the order of 10 4 — 10 6 times the masses of the primary condensa-
tions. Subsequently the relative velocities of the condensations were
determined by their gravitational interaction in close encounters and were
of the order of
v = Vl>F' (13)
However, in the aggregation of condensations traveling initially along
circular orbits, their angular momentum density with reference to the
center of the condensation is not conserved. Mutual attraction deflects them
from the circular orbits, and in collisions, depending on the initial difference
AR in distance from the Sun, they acquire either a negative or a positive
angular momentum; the change in angular momentum may even exceed the
condensation's angular momentum prior to collision (Safronov, 1960b). Thus
neither conservation of the angular momentum density of the aggregated
condensations nor reverse rotation can be inferred to be the inevitable
result of such aggregation— as often stated in discussions of Laplace's
theory. At first, aggregation takes place in a narrow zone along the orbit;
but due to encounters their relative velocities reach the values (13),
increasing with the mass. The zones that feed the condensations expand
61
correspondingly, well before all the material in the narrow zone along the
orbit has combined. In view of the continuity of the process of growth of the
masses and orbital eccentricities of the condensations, as well as the
absence of qualitative differences between the initial and subsequent stages
of growth (especially where the acquisition of angular momentum is concerned),
there is no justification for introducing the two concepts of "primary" and
"secondary" condensations.
If a condensation of mass m and radius r combines with another condensa-
tion of mass m' and radius r' , the latter will impart its own orbital angular
momentum K 2 with reference to the center of the condensation m and angular
momentum K 3 , related to the spin. The orbital angular momentum is
determined by the relative velocity v before impact and the impact parameter
flr:
K 2 = $rvm!. (14)
From (7), the angular momentum K 3 of the condensation, related to its spin,
is given by
K^~^m f r 12 =| |x siWriF'm', (15)
where fi is the inhomogeneity coefficient.
Since the plane of the relative orbit m' can be inclined in any direction,
the vector K 2 can have any direction. The direction of the vector K 3 is
correlated, if at all, with the direction of the vector of the total angular
momentum of the dust layer only at the beginning. After two or three
collisions, the direction of K 3 becomes random. In the process of aggrega-
tion, therefore, the vectors K 2 and K 3 add as random variables. Aside from
the randomly directed components K 2 and K 3 of the angular momentum,
during aggregation the systematic component K Y is also acquired; this is
related to the general revolution of the entire system of condensations
around the Sun and lies at right angles to the central plane of the system.
Qualitatively the component K x is of the same nature as the initial direct
rotation of a condensation formed from the diffuse material of the revolving
layer, which we estimated earlier with the aid of (9) and (10). It can also
be regarded as a special result of the asymmetry of the impacts among the
aggregating bodies (see Chapter 10). Then by analogy with (14) we can take
K 1 = ^rvm l . (16)
Assuming that the coefficient p does not depend on the mass of the growing
body, its approximate numerical value can be found from the present
rotation of the planets. We obtain p«?0.04.
If the masses m' combining with the condensation under consideration are
very small compared with m, the random variables K 2 and /C 3 will cancel
each other out almost completely and the angular momentum of the conden-
sation will be determined by its mean values as obtained by summing K v
Writing the relative velocities in the form v=Yj£, we obtain, by analogy
with (13) and according to (7.15),
62
dK, = i y/Gmr dm = -5L K ±1
and, consequently,
Alcorn*, ^ 1? j
where Pl =-_£-=-. Since ^ VS6 « 1 , Pi«0.1. For such a small exponent p, ,
contraction of the condensation should be rapid. Indeed, from (15), the
condensation radius
K 2
r cc — ^ oc m 2 ^ -3 ,
and its density
p oc -j oc m 10 " 6 ^*.
Therefore it is sufficient that the mass increases by one order of magnitude
for the condensation to contract to the state of a solid body with p^l.
A different result is obtained for the aggregation of condensations of
similar mass. The randomly oriented angular momenta K 2 and K 3 imparted
to each condensation are of the same order as the angular momentum of the
condensation under consideration and considerably larger than the mean
value K x , which can be disregarded. The resultant angular momentum is
then determined by the probable deviation from this mean, i. e.,
*' = 2 (*! + *»■ (18)
Here the direction of the vector K is arbitrary. Let us now evaluate K 2 and
K z . Let l be the maximum impact parameter for which aggregation of
colliding condensations still takes place. The mean value T 2 in the interval
between and l is given by
i
From the well-known relation in the two- body problem, we have the
following relation between the impact parameter l and the distance r at the
instant of closest approach:
72 r2 fi I 2G(m + ro') 1
Cm
For y 2 =— > where 6 is of the order of a few units, the first term on the
right is small compared with the second and can be disregarded. One may
assume that for r > r it would be difficult for the condensations to combine,
since the impacts are nearly tangential. Let us take r =fo, where p »l. Then
K\ = 7W 2 & G (m + m') m'V - 2p 2 K> ^ , (19)
63
where
In view of (15) one can write
where
(20')
P 3 =
2m*r"
The quantities p 2 and p 3 depend on the ratio m'Im. If we take some function
for the mass distribution of the condensations, we can find the mean values
of /J 2 and p 3 by simple integration. Then from (18) the mathematical expec-
tation of probable increase of the angular momentum of the condensation
with increasing mass can be written as
whence we obtain
JfTaomft+A. (21)
For the density of the condensation we derive
poom 1M ''-«p,. (22)
For the condensations to contract and not dissipate during aggregation, it is
therefore necessary that
/> 2 + />3<t. (23)
If the condensations are all identical this condition is not met (p 2 >l). On
the other hand, if the mass distribution of the condensations is such that
../ <m, then p % and p 3 become smaller than Pl and (17) holds for K. As we
have already seen, in this case aggregation of the condensations leads to
very rapid contraction.
It should be stressed that (21) merely gives the probable angular
momentum. Qualitatively the evolution of the condensations is the following.
Near-central impacts impart limited angular momentum and are accompanied
by rapid contraction: when two identical condensations combine, the density
increases by one whole order. Near- tangential impacts, on the contrary,
impart a large angular momentum and hinder contraction. Thus the disper-
sion of densities and sizes increases very rapidly from the start. Since the
unfavorable peripheral impacts have relatively little effect on the central
part of the condensation, one should expect a gradual increase in the concen-
tration of matter toward the center of the condensation. This means that
the central parts of most condensations would grow solid at a fairly early
stage, whereas the peripheral parts might remain in a diffuse state a fairly
64
m
long time. Transformation into solids takes place most rapidly in the case
of the most massive and densest condensations. The exponent for the latter
in (21) is small due to the fact that the effective m' is considerably smaller
than m, first because m is maximum, and second because they can easily
cut across diffuse condensations, acquiring from them only whatever
material is swept away directly by their cross section and imparts very
little angular momentum. It is sufficient to take m' = l f,m for condition (23)
to be met. In this case the conversion of condensations into solid bodies
will take place when the mass has increased by a factor of 10 2 at the Earth's
distance (10 3 at Jupiter's distance).
It is probable that the least massive condensations are unstable. For
these m'&m and, according to (19), p 2 ^6. The aggregation of such condensa-
tions will lead, on the average, to a reduction rather than an increase in
density. However, if a significant fraction of the cloud material remained
in the diffuse state, not all of it entering into the composition of the conden-
sations, then from (17) even the least massive condensations could have
contracted efficiently by assimilating this scattered material of low angular
momentum.
Taking a certain mean value p=p 2 +p s in expression (21) for the angular
momentum of the condensation, one can evaluate the duration of the conden-
sation's evolution from the ordinary growth formula
dm J2 _ 2 19 2ic o 9 / A . 2Gm'
dm -72-., ~ 2 19. 27C ° <l{a i 2Gm\
Next, setting Kccmf, we have rocm 2 ^ 3 . For u = yjGmfir and in view of (12) we
obtain
mM^^Sai+iflg,*^*. {24)
The layer thickness H is related to v^sJ^RTfa by (5.17). The gravitational
attraction of a single cluster would give H oc v 2 , and that of the Sun H oc v. At
the start, when the density of the cluster is nearly critical, H oz v 1 - 1 . When m,
and correspondingly!;, increases several times (which takes place within
fewer than a hundred revolutions), H increases significantly and the z
component of solar gravitation increases. Quantity H then becomes nearly
proportional to v and, from (3.5), is given by Pv/4, where P is the period of
revolution around the Sun. Integrating (24) and introducing m /rg^4a, in
accordance with (6), we obtain the time required for the condensation mass
to increase from m lt at which the relation (3.5) becomes applicable, to m:
4«4 (7 — 4j»)(l+26) [(^;) P ~{^ P ] P - (25)
Thus the time during which the mass increases from m to m proves to be of
the order of (mim o y~*pP t Since poem 10 - 6 *, the time during which the condensa-
tions transform into solid bodies will be of the order of
i-*p
T^^'P, (26)
65
where p 1 is the condensation's density after initial contraction, i. e., roughly
one order greater than the Roche density. For m' = m/4, the case discussed
above, one obtains pml.2 and the time in which the condensations convert
into solids proves to be of the order of 10 4 years at the distance of the Earth
and 10 6 years at the distance of Jupiter. The time required for evolution and
transformation into solids may vary widely from one condensation to another,
but on the whole the entire system of condensations converted within a
cosmogonically short time into a cluster of solid bodies. The formation of
numerous bodies hastened the break-up of condensations lagging behind in
their development. Among the planets of the Earth group, the condensations
transformed into solid bodies much sooner and with much smaller masses,
on the average, than in the region of giant planets.
66
CONCLUSIONS
In early works by Shmidt (prior to 1950) it was assumed that solid bodies
large enough to hold the particles falling into them were present in the proto-
planetary cloud from the beginning. In an important contribution to the
advance of the study of the early evolution of the protoplanetary cloud,
Gurevich and Lebedinskii demonstrated that the cluster of solid bodies was
formed as a result of the flattening of the dust layer enveloping the Sun and
its disintegration into numerous condensations.
In Chapters 2 and 3 it was shown that the dust layer supposed by Gurevich
and Lebedinskii to have constituted the primordial state is the natural result
of the evolution of a gas- and- dust cloud of cosmic composition revolving
around the Sun. It was found that the protoplanetary cloud was stable under
small perturbations. The emergence of convection inside the cloud would
require a very steep temperature gradient, such as could not have been
achieved inside it. Calculation of the gravitational energy released by the
cloud when interaction among turbulent eddies caused them to move closer
to the Sun revealed that this energy was not large enough to maintain
turbulence inside the cloud. Primordial random macroscopic motions
present in the cloud must therefore have died away rapidly. This caused
the solid material to separate out from the gaseous material. Dust particles
began to settle toward the central plane of the cloud, forming there a layer
of high density. The break-up of the dust layer was examined in detail in
Chapter 5 on the basis of the theory of gravitational instability. The aggre-
gation of the numerous dust condensations formed as a result of the break-up
of the layer (see Chapter 6) led to the formation of a cluster of solid bodies.
The subsequent evolution of the cluster and the formation inside it of the
planets are discussed in Part II of this book. In Chapter 3 it was shown that
gravitational instability was probably not present in the dust layer in the
portion of the cloud adjacent to the Sun; this is because the high degree of
flattening of the layer necessary for its presence could not have been
achieved owing to the perturbation entering this zone from the evolving
active Sun. In this zone the growth of solid bodies must have resulted from
the aggregation of particles in collisions. Analysis of temperature conditions
in the protoplanetary cloud (Chapter 4) indicated that hydrogen could not
have been present in the solid state in the region of giant planets and that
the planets rich in free hydrogen— Jupiter and Saturn— must have acquired
it in the gaseous form by accretion in the concluding phases of growth.
Thus the principal stages in the evolution of the protoplanetary cloud
enveloping the Sun are becoming increasingly clear and precise. The
problem of the origin of the cloud itself, however, is still unsolved. From
67
our review in Chapter 1 of present-day theories regarding its origin, it is
seen that they are all beset by considerable difficulties. The most promising
theories at present seem to be those that envisage a common formation of
Sun and protoplanetary cloud.
The urgent tasks today are to construct a consistent picture of the forma-
tion of the protoplanetary cloud enveloping the Sun and to study its physico-
chemical evolution.
68
Part II
ACCUMULATION OF THE EARTH AND PLANETS
Chapter 7
VELOCITY DISPERSION IN A ROTATING SYSTEM OF
GRAVITATING BODIES WITH INELASTIC COLLISIONS
20. Velocity dispersion in a system of
solid bodies of equal mass
The process of planetary accumulation consists mainly of the collisions
among and aggregation of numerous protoplanetary bodies. The relative
velocity of these bodies is one of its major characteristics, since it deter-
mines the rate of planetary growth and the degree of fragmentation of the
colliding bodies. There is a close relation between the relative velocities
of the bodies and their size distribution. Initially the bodies, formed inside
a flat dust disk, moved along nearly circular orbits and had low relative
velocities. With aggregation and increasing mass, however, their gravita-
tional interaction increased, as did the relative velocities and, correspon-
dingly, the oi'bital eccentricities.
An approximate expression for the velocity dispersion in a system of
protoplanetary bodies of equal mass was obtained by Gurevich and Lebedin-
skii (1950). The time of encounter of the bodies is many times smaller than
the time required for one revolution around the Sun. Encounter can there-
fore be treated as in the two- body problem: the relative velocity vector v of
approaching bodies of mass m does not change in magnitude but merely turns
through the angle ty&GmfDv 2 , where ijj-^n/2. In the process a change takes
place in the eccentricity e of the body's orbit, given roughly by the expression
Dv y \±)
where V e is the circular velocity and D the impact parameter. Next the
authors assumed that for very close encounters where D = 2r (r being the
radius of the body), the orbital eccentricity increment Ae is of the same
order of magnitude as e itself. One can then find e from (1), and conse-
quently the relative velocities
* = #.~V%.
(2)
The result is correct in substance, but the expression for v needs to be
refined. Relation (1) holds only for small ^, i.e., for "distant" encounters.
Since — ^'-^-!^r|), it should follow that Ae? < e. But the authors applied
69
expression (1) to close encounters, taking D = 2r and presupposing that Aet&e.
It is not clear what degree of error this produces, as for large -ty the relations
become more complicated and fail to yield an expression similar to (2) for v.
In a system with differential rotation the dispersion of velocities of the
gravitating bodies is caused by the conversion of the energy of ordered
motion into energy of random motion. The former is renewed in turn by
the potential energy of the system with reference to the central mass — the
system is somewhat compressed at right angles to the axis of rotation. If
the collisions between the bodies were absolutely elastic, their velocities
would increase steadily and no relation of the type (2) could hold. In a real
system with inelastic collisions the dispersion of velocities is determined
by the balance between the energy acquired in encounters and the energy
lost in collisions. The assumption that Ae^e for D = 2r (Gurevich and Lebedin-
skii) is essentially an implicit expression of this balance. The important
"characteristic" dimension should indeed be of the order of 2r. However,
expression (2) does not tell us how the velocity dispersion depends on the
nature of the collisions and on the degree of their inelasticity. The author has
therefore carried out a more detailed analysis of the problem with a view to
obtaining this dependence in an explicit form (Safronov, 1962d).
a) Dispersion of velocities for small mean free paths. Consider a rotating
system of identical bodies of mass m and radius r not containing any gas. As
long as the mean free path of the bodies is short compared with the distance
from the Sun (i. e., the bodies themselves are small), the increase in velocity
dispersion can be estimated from the ordinary hydrodynamic formulas for
the dissipation of the energy of mechanical motion of a fluid due to viscosity.
In an axially symmetric flow with angular velocity ^ (R), the amount of ener-
gy dissipating per cm 3 per sec due to molecular viscosity is given by the
expression (Lamb, 1932)
*=^y=if^(£y. ^
where t^I/3 pyA, is the coefficient of viscosity and R the distance from the
axis of rotation. Applied to the system under consideration, E is the amount
of energy of ordered (rotational) motion of the system that converts into
energy of random motion. The above relation can be given a simple physical
interpretation. On the average one third of all the particles move in a radial
direction. Within the mean free time t, particles traversing the mean free
path X acquire the relative velocity of differential motion Av = r(~\\, which
changes from ordered to chaotic. The thermal energy —fo*=—R 2 (JL\ )? is
generated per unit mass within the timex = X/y, Since \- = 2X 2 = 2X 2 , dividing
this expression by x and multiplying by p/3 (where p is the density of the
medium) we recover (3).
There is little gravitational interaction between small bodies. But if the
relative velocities are also small, the gravitational attraction of the bodies
will need to be taken into account. Let x § be the time between two successive
collisions of a body with other bodies and x ff the time between successive
close encounters involving substantial energy transfer. This occurs when
the relative velocity vector of the body turns through an angle — n/2.
70
From (3), elastic collisions or close encounters will cause each body to
acquire an average energy of relative motion i? 2 XV 2 /3i ? per sec per unit mass.
Inelastic collisions are less effective in this respect. When colliding bodies
aggregate, the relative velocity vector deflects from its initial direction by
only about ji/4 on the average. We can therefore assume that the increase
in the energy of relative motion during collisions among bodies amounts to
C^XV 2 ^-:, per sec per unit mass, where Ci<l. Then
-=**=iG+£) m, (£y. (4)
and the mean free time t within which v turns through ji/2 is given by
±=± + K (5)
On the other hand, over each time interval t s the body will lose part of its
own energy due to inelastic collision. Let the energy e 2 x s lost per unit mass
amount to the fraction C of the kinetic energy of the body at impact. Let us
denote by v x the velocity of the body (relative to the circular velocity) after
a previous collision and by v % its velocity before the next collision. Then
i>J = i>J + 2e lV 2 Vt = &i (6)
If v* is the mean of v\ and v\, then
v.=f(«* + vJ. (7)
Here v denotes the root mean square velocity. We take v 2 =~v 2 as for the
Maxwellian distribution. As a result of the combined action of both effects,
the body acquires the following amount of energy per unit mass per second:
The geometrical collision cross- section of two bodies of radius r is 4nr 2 . But
due to the low efficiency of near-tangential collisions, it is actually smaller,
and we shall designate it by b*r s . Due to gravitational attraction the collision
cross-section increases ( 1 + 2 G/n/K 2 r) times, where V is the relative velocity
of the colliding bodies before the encounter. It can be assumed that on the
average V 2 = v 2 -\-v\. From (6) and (7) we obtain
(9)
"J — " 2 — AC'
where
A = ei /e,- (10)
Therefore the mean free time between two successive collisions is given by
45r (»e/G) v »
71
(11)
where for convenience we have set
rrKk= a - (13)
The time x p between encounters can be assumed to be equal to the relaxation
time T s or T Dl after Chandrasekhar (1942),
T »r I== ±i/I < yi )' / ' - w g >' /a (14)
' 16r i G2mp In (i + D Q vt/2Gm) "gp^Mn (1 + Z> /2«r) '
where Z?„ is the mean distance between the bodies. Consequently,
x, _ , 92 In (i + D /29r) (15)
*p S (1 + «9) *
Quantity D can be expressed in terms of the number n of bodies per unit
volume (Chandrasekhar, 1943):
o «*O.554n-v.. (16)
According to (3.5)
Therefore
P»=£. (17)
n = el m =-L- (18)
Substituting for v from (12) and carrying out some simple operations, we
obtain
^~fWf- (19)
Eliminating the density p from (11) with the aid of (17), we find that
46r p (20)
3 \fl £a (1 + a6) 4
The condition that the foregoing relations be applicable for e (short mean
free paths) can be written as t,<P/4. This gives r<£a(l + a6)/8 ~~ 10 cm for
the terrestrial zone. On the other hand, the cloud of particles which we are
investigating cannot be in a state approaching gravitational instability. This
imposes the condition that the relative particle velocities v > 10 cm/sec.
Then, from (12), it is necessary that 6< 10~ 7 . Introducing the foregoing
expressions for z s and t/c, in (8) and setting 8< 1, we obtain
C«*/ 4*(2-C)Wr» ,\ (21)
72
In the terrestrial zone e = for r=r c & 3— 4 cm. If r < r e , then e < and the
particle velocities decrease. During collisions particles aggregate and their
size increases. Depending on the initial density of the cloud and the initial
ratio rir c> either gravitational instability will develop inside the cloud or,
before this can happen, r will increase to r es the particle velocities will
begin to rise, and the onset of instability becomes impossible. The particle
velocities and therefore also the uniform thickness of the layer decrease
roughly in inverse proportion to the size of the particles. Therefore if
initially rjr c <p/p cr m order of magnitude, then gravitational instability will
develop before r increases to r e . If r^>r e , then e>0 and the particle veloc-
ities will increase. Their mean free path increases in the process and within
a few periods of revolution the initial equation (3), and therefore the expres-
sions for e derived from it, ceases to be valid.
b) Dispersion of velocities for large mean free paths. There exists no
satisfactory theory of the transport of matter and motion in revolving
systems for large mean free paths. In Chapter 2 we noted that in Prandtl's
semiempirical theory, which was developed for turbulent motion (mixing
length comparable with the dimensions of the system), it is assumed that the
shearing stresses are determined not by the angular velocity gradient (2.17)
but by the angular momentum gradient (2,18). For Kepler rotation (o>oc R-*i*
and mR 2 oc R 1 !*), this causes a threefold reduction in the shearing stresses and
a ninefold reduction in the generated energy compared with expression (3).
Since Prandtl's theory is also nonrigorous, we introduced the additional
factor p' < 1 in the expression equivalent to (3).
Introducing p' in (3) and replacing X 2 by \^R 2 > we can write the expression
for the energy of relative motion per unit mass acquired by the body within
the mean free time as follows:
v -*r*(£fm. (22)
Here AS 2 is the mean value of the square of the radial displacement of the
body. For short mean free paths, AS* — X r =2X 2 . For large X the quantity A/F
can be much smaller than X 2 ", as the maximum radial deviation A/? m does not
depend on X and is determined only by the orbital eccentricity (&R m ~~eR). Let
us evaluate Ai? 2 . For small orbital eccentricities e,
R f — R=—*!. JJ«a(l— ecoB?) — rt. ( 23 )
1-fecosf v T/
For large X the true anomaly q> during encounter can assume any value be-
tween and 2jtwith nearly equal probability.
For bodies whose relative velocity is directed radially at the initial instant
(v = vr), a = /? and
2*
/?' — #« — R e coscp, AA^fi'e*—- j cos«<pd<p = i- *V. (24)
o
For bodies with relative velocity along the orbit (v = v v ), a~Rf(\ ± e) and
73
R 1 — R& -Re(±\ + cos<p),
Z&1VR&-L \ (±1 + COS <p)2rf ? =|/?V. ( 25 )
For small mean free paths, bodies with relative velocities along the radius
(v = v R ) contribute to e, (expression (5)). For large X bodies with relative
velocities along the orbit (v = v 9 ) contribute three times as much to e 1# Taking
the role of both groups into account, we can substitute 2 R 2 e 2 in (22) for AR* t
in accordance with (24) and (25). As will be shown below (see (43)),
i**eF„ v f n*±V„ (26)
where V, = ioR is the circular velocity.
Assuming that the mean inclinations of the orbits are equal to their mean
eccentricities, as is the case in the asteroid system, we obtain
«*=!* (27)
We will therefore take the mean square relative velocity to be
■M-w+^+^l^:- (28)
Introducing &R 2 = 2R 2 e 2 and to = \JGM/R 3 in (22), we obtain
The energy e 2 lost in collision can be determined from (7). In view of (5),
the change in the energy of relative motion of the bodies per gram per second
is given by
.=,_, = ,( 1 _c ) _g = c, [ « i ^: (:t+Ci) _ l] . (30)
Introducing the expression for zjx from (15), we obtain
<=m^n m \v + T r) +^}- i i (3i)
In a system in which the bodies collide but do not aggregate (constant m and
r), their relative velocities should tend to a certain "equilibrium" value
which can be determined from the condition that « =0; this gives
ft2 _ UC"C 1 (2-C)P"](i+«8) (32)
— 4(2-C)P"ln(i+Z> /2er) "
For smaller velocities the parameter 9, which is related to v as in (12), is
greater than the equilibrium value (32), and therefore e> and v should
increase. If the velocity is greater than the equilibrium value, 6 will be
smaller than (32) and e< 0. In this case the velocities of the bodies should
decrease.
74
For j3* = 0.2 (see below), expression (32) will give ~1. For C< 2 ^ g
» 0.2 the expression for is found to be imaginary. In this case the loss of
energy of motion of the bodies in collisions does not compensate for the cor-
responding increase caused by gravitational interactions in the rotating
system and the velocities of the bodies will increase indefinitely.
From the standpoint of the process of planetary accumulation, it is more
interesting to study systems in which the bodies aggregate in collisions and
whose masses increase. Let us suppose for simplicity that aggregation
takes place in every collision and that the masses of all the bodies remain
the same during the growth process. As before, we will seek to obtain an
expression for v in the form (12). Then, differentiating with respect to the
time for constant 0, we obtain
l G dm
(33)
The mass of the body will double within the time t # :
£*. = «■ 04)
Consequently,
l__Gm____t^__C£ f (2-Qp» f 462ln(l + J /26r) , r f\ 1 ,~ ,
For this value of e,
From (35) we obtain the following expression for 9:
e2 2 + 3C-3p»<2-C)C, e (37)
Quantity depends very weakly (logarithmically) on the ratio D /r which, in
turn, depends weakly (as the cube root) on r. Relation (12) is therefore a
physically valid expression of the dependence of the relative velocities of the
bodies on their masses and radii. The quantity in this expression can be
regarded as practically constant. Relation (12) can be seen as a generaliza-
tion of formula (2) of Gurevich and Lebedinskii, as it constains the parameter
0, which is dependent on the properties of the system of bodies under
consideration.
When bodies combine in collisions, it is not difficult to evaluate the
parameter C characterizing the degree of inelasticity of the collisions. Let
two bodies, each of mass m, have the same velocity v (in magnitude) of
relatively Keplerian circular motion and let c|> be the angle between the
vectors v. The velocity v' after collision and aggregation can be determined
from the condition that the angular momentum is conserved:
2nw ! = 2mv cos y , i/ = v cos -j . (38)
75
From the definition of C,
0; s = i;« — w'» and C = sin 2 A.
(39)
The collision frequency v is proportional to the relative velocity V of the
bodies and to the effective collision cross- section I 2 :
v(<j>)ocFZ 2 oc2ysini/H
2Gm
r4i>2 sin 2
T)
OC Sin-j-| ;
2 sin -
(40)
For a random distribution of the vectors v over the directions, the mean
value of C is
._!<>">_*'
it
sin2-
\
2sin-
sin tydty
JvdQ
6 + ai
I
sin ~2 + *
2 sin
f)*
10-f- 156*
(41)
J sin <|><ty
Thus C in turn depends on 8, though relatively weakly: for 8 = 1, C = 0.44 and
for 8=3, C= 0.38. Without introducing a major error, one can assume C =
= 0.4 in expression (37) for 8.
In reality the velocity distribution of the bodies is not isotropic. Velocities
in the radial direction are, on the average, twice the tangential velocities.
However, this does not alter C substantially.
Earlier in the expression for t we introduced the quantity d < 1> which is
linked to the fact that in inelastic collisions the velocity vector of a body will
turn through an angle smaller than n/2 on the average. The relaxation time
T p} according to Chandrasekhar, is defined as the time in which the sum of
the squares of sines of the angles of deflection of the body in encounters
reaches unity. In the case we are considering, in which colliding bodies
aggregate, v is turned through the angle (Ji/2. Therefore
C 1 = sin 2 -| = C.
(42)
Tables 6 lists the values of the parameter 8 calculated from (37) for C=C a =
= 0.4, 5 = 2, fl- 2 andp'= 0.2 for the Earth zone (<7~10g/cm 2 ) and for the
Jupiter zone (cr = 20g/cm
densities fl.
as a function of the radii r of the bodies and their
TABLE 6
e
6, g/cm 3
, LIIl
Earth zone
Jupiter zone
10 2
2
1.4
1.1
10 3
2
1.08
0.96
10 4
2
0.94
0.84
10 5
2.5
0.83
0.77
10 7
3.0
0.70
0.66
76
For £ = 4 the values of 8 come closer to 6=2, as adopted in expression (2) of
Gurevich and Lebedinskii; but for fairly large bodies they are still percep-
tibly less.
21. Increase in energy of relative motion
in encounters
In all the preceding calculations a significant error stemmed from the
uncertainty in the parameter 3' in (22), which was introduced in an attempt
to draw an analogy between rotational motion for large mean free paths and
turbulent rotational motion. As the theory of turbulence has so far failed to
give a definite numerical value for this parameter, it is desirable to attempt
to evaluate jS' by another, more direct method.
In order to avoid complicating the problem excessively, let us consider
the following idealized scheme. The body m x is traveling along an elliptical
orbit of small eccentricity e in the central plane of a rotating system (per-
pendicular to the axis of rotation). Let the other bodies m 2 which it encoun-
ters move in the same plane along circular Kepler orbits to which they are,
as it were, fastened, not deviating from them during encounters. This
assumption indicates a kind of averaging of the results of the encounter of
the body m l with other bodies of different velocities with respect to the
circular velocity but directed with equal probabilities along different
directions, so that the mean value of this relative velocity is zero. Since
during encounters the velocity vector v lg of the body m x shifts with respect
to the center of gravity of m x and m 2 , the energy transferred will depend first
and foremost on the magnitude of this velocity. The assumption that the
relative velocity v % of the bodies m a is zero reduces v lff , but the coincidence
of the center of gravity with m a increases v lf , so that the errors introduced
by these two simplifications compensate each other to a large extent. From
(29) it is evident that the expression V s pV represents the mean increase in
the relative energy of the body due to encounters within the relaxation time,
i. e., for an average rotation of 90° in the direction of the relative velocity.
We will therefore consider encounters of m 1 with m 2 at different points on the
orbit of m x for which the vector v lg =v 1 rotates by 90°. We assume further
that all the bodies have identical masses: m 1 =m % —m.
It can be sho#n that at the point of intersection between the elliptical
orbit (semimajor axis a and eccentricity c) and the circular orbit (radius R),
the relative velocity of the body on the elliptical orbit with respect to the one
on the circular orbit is given by
»? = "?[3-£-2/£(l-e')j. (43)
Introducing
11
we obtain
= e 2 F 2 1 — -^-cos 2 tp — e(cos<p — ^- cos 2 <pj+ . . . 1.
(45)
Up to first order infinitesimals in e, v x = v t&-?eV 9 at perihelion and
aphelion (true anomaly <p = and 180°), and v 1 =v R ^eV c at the intermediate
distance B = a {<p= 90 and 270°), i. e., it becomes twice as large.
e>*2e
vr'W*
FIGURE 3. Change in orbital eccentricity of a body due to
close encounters with other bodies at different points along
the orbit. When the relative velocity vector V! rotates
through the angle 7r/2 without change of magnitude at peri-
helion and aphelion of the initial orbit, the eccentricity of
the new orbit is half the initial value; when this takes place
at the intermediate distance, it becomes twice the initial
value.
The difference between the relative velocities v x at different points along
the orbit is the main factor controlling the redistribution of velocities and
energy during encounters. Bodies lying at perihelion or aphelion of their
orbits (v=v 9 ) will, after close encounter and rotation of v 1 by 90°, lie at the
intermediate distance along the new orbit (v = v R , Figure 3). But as the
quantity w, remains the same, the orbital eccentricity decreases by a factor
of two. If, on the other hand, encounter takes place at the intermediate
distance (v = v K ), the 90° rotation in v x will be accompanied by a twofold
increase in the orbital eccentricity. The energy of relative motion should
be determined not by the value of v\ at the given point but by the mean value
v\ along the orbit:
^(i-i^i^)=| eV ;,
(46)
78
i. e., by the square of the eccentricity, when e is small and higher order
terms can be disregarded. Therefore the energy of relative motion will
decrease by a factor of four in encounters at perihelion and aphelion and
increase by a factor of four at the intermediate distance. On the average
this energy increases, since — f^-yj-|-4i;5J=-g-i^>yJ. This indeed constitutes
the physical essence of the mechanism by which the dispersion in the veloc-
ities of bodies in a rotating system increases due to their gravitational
interaction for large mean free paths.
Let us evaluate the increment in the energy of relative motion. By
examining the two- body problem one can derive the following relations
between the parameters R and V e of the circular orbit, the parameters a and
e of the elliptical orbit, the relative velocity v x at the point of intersection
between them, which lies at the angular distance <p from perihelion, and the
angle i|> between v x and V c :
«•= 1-5.(1 + ±*mtf, (47)
5-=l_2£cost--£, (48)
it sm +=r+7^ (49)
Then correct to the infinitesimals e 2 and v\jV]
e 2^il(l +3cos 2 <|>). (50)
After encounter, v[ = v v t|>' = t|> + y , and therefore
A e 2 == ^_ e 2 = Mj^cos 2 (t± y)-cos 2 f]= 3^-(2sin 2 t - 1). (51)
From (45) and (49) we obtain
Ae 2 ^3* 2 (l-A cos 2cp).
52)
In the two-dimensional problem under consideration, the frequency of
close encounters v(tp) is proportional to DV, where V = v x is the relative
velocity of the approaching bodies and D the impact parameter for which the
vector v, will rotate by ir/2. Then DcoV~ 2 and from (45)
(53)
v(<p)oc l/^oo^l— T cos 2 <p) \
The mean value of Ae* for all points of the orbit is given by
* $ (i - T cos*?) (1 - x cos2 ?)~ l/ '
te*= v( ? )Ae 2 ^ = 3e 2 ^ = 0.8e 2 . (54)
° S v 1 ~ T cos v * df
79
For the spatial problem one should take v(y)<x,D 2 V and
K« 1/^ + ^ = ^1/1(1 +|sin« ? ). (55)
In this case Ae 2 ^ 0.69e 2 , i. e., it differs only slightly from the above value.
From (29) and (46),
and
p-^?~ - 3 - (56)
The result obtained applies to the motion of m x in a plane perpendicular to
the axis of rotation, where encounters will alter the orbital eccentricity
most effectively. In the general case of motion along inclined orbits, one
should expect a smaller value of £'. For purposes of numerical evaluation
we will assume that j3' — 0.2.
We have confined ourselves here to close encounters (ty l — ( P = yY But it
is known that distant encounters play no less important a role in the exchange
of energy. As v x rotates through the angle A\|), the value of e 2 determined
from (50) will change as follows:
v* v 2
Ae 2 = 3-+[cos 2 (<J> + A<|>) — cos 2 t|>] = 3-|-[sin 2 A^(sin 2 <|» — cos 2 1|») —
V c ^e
— 2 sin A<|> cos A^ sin i|* cos <[>]. (57)
Replacing ip by <p in accordance with (49) we obtain
Ae 2 = 3e 2 rsin 2 A<M sin 2 <p — -j- cos 2 f J— sinAt|>cosA<f sin 9 cos?]. (58)
In view of the symmetry of v (9) (according to (53) and (55), v (<p) is an even
function of<p), when averaging over <p the term with sincp cos <p gives zero. The
factor containing sin Ai|> represents the increment A« 2 in close encounter
( A\|>=jt/2), as given in (52). Therefore
A? = 81^(5?^. (59)
Hence it is seen that the effect of many distant encounters will be the same
as that of one close encounter if 2 sin 2 At|* ^= 1 . But this is the condition that
defines the relaxation time T D , after Chandrasekhar, which is nearly the
same as the expression T E taken above for the time x ff between encounters
(see (14)). Thus when distant encounters are allowed for, the situation
becomes quite satisfactory.
80
22. Velocity dispersion of bodies moving in a gas
In view of the fact that the gaseous component of the protoplanetary cloud
(amounting originally to 99% of the mass) was not scattered at once, in the
early phases of growth of the protoplanetary bodies the gas offered resistance
to their motion. Hence there is a need to adjust our foregoing estimate of
the velocities of the protoplanetary bodies.
The importance of friction in the gas is immediately apparent when one
compares this effect with the deceleration which occurs during aggregation.
Let e 3 be the energy lost by a body due to the resistance of the gas per gram
per second. Then, from (3.1),
Wr
and, from (29) and (6),
= Fv = cv 2 , c = V (60)
pv* _Cu| (61)
e 2 — 2x t *
T- »
From (11), (17) and (60) we have
t 3 _ 2cz t v* __ 8 v*° 9 ° g / 62 \
H lv\ 3\/2~£C(l +a6) v\ 9 p (l+a8)o p *
At the initial stage of growth ojo p ~lQ 2 and the deceleration due to friction is
far more effective than that due to inelastic collisions. It is only when the
mass of gas inside the cloud becomes less than the mass of solid material
due to dissipation that the resistance of the gas can be disregarded.
From (60), (61), (9) and (35),
» = «i— ,— »=(^- Tf(2 i K) -cy=S;^
(63)
where, as before, &=e/e a + 1 and is expressed in terms of C in the form (36).
Let us determine xjx from this and compare it with its expression from (5)
and (15):
, t, - 462ln(l + Po/2er) _ C x, 2 + (3-*)C (64)
(, i" 1 "T, —,, *"r 5(1+ aft) ~ P "T 3pt'(2-K) *
Eliminating cx 9 as in (62) and denoting the parameter 6 for motion of the
bodies in the gas by 6 ? , we obtain
6 .-3^1n(l+i? /2e,r)[^2- V + 4(2-K) 6(1+«VJ- V }
The values of 9, are cited in Table 7. The density 6 of the bodies is assumed
to increase with their size from 2 to 3 g/cm .
When the bodies are small (a few meters in diameter), 6, is an order of
magnitude greater than 6 as computed in the absence of a gas. But even in
a system of larger bodies (tens of kilometers in diameter) when the gas is
retained 6, will exceed 6 by a factor of 4 to 5. The rate of growth of the
protoplanetary bodies is proportional to (1+2 9)-. The presence of 'the gas
81
TABLE 7
r, cm
Planet
a p, g/cm 2
U/'P
10'
10 3
10'
10 s
10 7
Earth . . .
10
200
43
22
15
12
9.1
"
10
70
16
10
7.8
6.4
5.1
Jupiter . .
20
70
10
8.0
6.7
5.7
4.8
Neptune . .
1
70
6.4
5.6
5.1
4.7
4.1
must therefore have substantially hastened the growth of the bodies. It also
substantially reduced the effectiveness of fragmentation in collisions.
23. Velocity dispersion in a system
of bodies of varying mass
When evaluating the velocity dispersion in a system of bodies of varying
mass one meets additional difficulties stemming not only from the increasing
complexity of the formulas but also from the necessity of knowing the size
distribution function of the bodies. The latter must be determined from a
complicated integrodifferential equation (see Chapter 8) whose solution, in
turn, depends on knowledge of the velocities of the colliding bodies. The two
problems should strictly speaking be solved simultaneously. But the complete
problem is insoluble in analytic form and has to be split into two problems,
one in which the distribution function of the bodies is assumed to be known
and the other in which their relative velocities are assumed to be known.
In this book we will confine ourselves to an approximate estimation of the
main factors governing the velocities of the bodies in the system, presuppos-
ing for simplicity that the size (mass) distribution of the bodies obeys a
power law:
n (m) dm = cm q dm, c = (2 — q) pmp 2 ,
(66)
where m x is the mass of the largest body in the distribution. The above
expression for c is suitable for q < 2, which corresponds to p=Sq -2< 4 in
the distribution n (r) = c l r~ p over the radii.
We assume as before that a body of mass m' loses energy e 2
'2t,
per
second due to collisions. We determine z 8 as the time within which the body
m' collides with bodies having total mass m' and, combining with them,
doubles its mass. Consider the case p > 3 (i. e., q > 5/3) where t, is less
than the "lifetime" of m\ i. e., than the time between collisions among m'
and larger bodies. Then the doubling of m' will take place thanks to the
aggregation of smaller bodies m" < m'. Since the shift in the direction of the
velocity of m' in collisions with small bodies m'is small, C x will be small and
x?xx r Frequent distant encounters and frequent collisions with smaller
82
bodies will cause a uniform energy generation e 1 and energy absorption e 2 ,
so that the velocity v can be regarded as constant over the time x t . There-
fore
e 1 T, = pV 1 6^ = ^/2. (67)
Then, in view of (35),
•=•.-■>=(£-£)'=£. (68)
where t, is the doubling time of the mass m of the largest body. Hence
For a power law of mass distribution
V^M™'/™)"" 5 ' 3 .
Since the ratio t,/t # is a function of 0, the latter can be determined from
expression (69).
In Chapter 11 we will show that the masses of the bodies that fell into the
Earth did not exceed 10" 3 Earth masses. This means that in the concluding
phases of growth the embryo Earth was far in advance of the other bodies
and that it dropped out of the general mass distribution cm~ q of the bodies.
Therefore two cases are to be considered: a) the initial stage of accumula-
tion in which the planet embryo m is the largest body in the distribution m~ q ,
and b) the concluding stage in which the largest body m x in the distribution
m~ q amounts to a small fraction (~lCf 3 ) of the mass of the planet embryo.
a) Initial stage. According to Chandrasekhar (1942), for a body of mass
m' moving with velocity v in a system of bodies of mass m" , the relaxation
time T p is given by
T ' = , ", p* y (70)
where n is the number of bodies m" per unit volume and ffl (x Q ) is a function
of the ratio of the velocity v of the body m' to the velocity dispersion of the
bodies m".
To obtain the relaxation time T' D for a body m! moving in a system of
bodies of varying mass, it is necessary to take the inverse of T D (according
to (70)) and integrate over all m":
^"^(i+g-^tf-j)! ■rt. M -W. (71)
Gm
Setting m 1 ~m and v 2 =jrp and taking n(m) in the form (66), we obtain
r 3-g ^ (72)
B 2 — q STtf^mp *
83
where we assume that f x = M In ( 1 4- _ , ,° v — -V
\ ' G(m' + m")/'
The time t, within which the body m' collides with bodies m" < m' of total
mass ro' can be found by integrating the inverse of x t , given by expression
(11), over m" ' :
_ = il_ I m"n (m!) dm" = — iSpuf— ) f
(73)
where
f>=V^^+ 2 Prm-
Introducing the values of T D (instead of t ) and x 9 in (69) and assuming that
Gm
v 1 - =£^, we obtain
r
T , 3 — 9/2 \ ^ /
(74)
and
e i2_3-gfr/ t /m \* 3
— 2-98^ U'i
(75)
where & is defined by (69).
The values of 6' calculated from (75) for the Earth zone for C = 0.7 are
listed in Table 8.
table 8
p=3
P = 3.b
r, cm
r' = r
r' =0.1r
r' =0.01 r
r' = r
r' -0.1 r
r' = 0.01 r
e
105
10'
3.5
2.3
1.9
1.2
1.1
09
1.2
1.0
0.9
6.4
4.0
2.9
2.6
2.2
2.0
5
4
3
1.5
1.1
0.9
The first column shows the radius r of the largest body in the planet's
zone. The next columns list the values of 8' for bodies of radius r\ For
p=~ 3, the smaller bodies have a smaller 0' (greater velocities). This is due
to the relative decline in importance of collisions resulting from reduced
gravitational focusing and the conversion of the collision cross- section into
a geometrical cross-section. A similar effect is observed for p= 3.5, though
only when r moves to 0.1 r. For smaller r' the parameter 9 again increases
84
due to the high collision frequency, which increases progressively with
decreasing r'. As the radius r of the largest body (and correspondingly
of all other bodies) increases, 0' decreases: owing to the rising velocities
the thickness of the system increases, with DJr and j x increasing corre-
spondingly. The values of for a system of identical bodies of radius r
are given for comparison in the last column. The relative velocities (in
cm/sec) of the bodies corresponding to the values of 6' are given in Table 9.
TABLE 9
P :
= 3
p =
: 3.5
r, cm
r' — r
r'<r
r' = r
r'4r
103
0.4
0.7
0.3
0.3
105
50
80
40
40
107
6600
9600
5400
5300
In the early evolutionary phase of the system relative velocities were
small, so that combinations were predominant in collisions, as is seen from
the table. Fragmentation became significant only when the largest bodies
grew to a diameter of several kilometers. Doubt has often been expressed
in the cosmogonic literature as to whether the aggregation of small rigid
bodies was possible. Thus in a review by Gold (1963) the formation of
sufficiently large bodies capable of further growth by gravitational attraction
is included among obscure and delicate problems of planetary cosmogony
urgently requiring a solution. One way of approaching this particular problem
was to develop a theory in which gravitational instability of the dust com-
ponent inside the cloud led to the formation of sufficiently massive dust
condensations that evolved eventually into bodies. The above evaluation of
the relative velocities of the bodies shows that when conditions necessary
for gravitational instability were absent, the bodies could have grown to a
diameter of several kilometers by direct aggregation in collisions. Doubts
as to the possibility of direct growth of the bodies at the early stage stem
exclusively from the absence of quantitative estimates of the relative veloc-
ities of the bodies and from notions inherited from von Weizsacker (though
not justified in any way) regarding the prevalence of velocities of the order
of 1 km/sec in the protoplanetary cloud.
b) Concluding stage, m>m 1 . When the mass m of the planet embryo is of
the same order as the mass of all the bodies in its zone, perturbations of the
embryo m in the motions of other bodies become significant and it becomes
necessary to consider these separately when evaluating the relaxation time x*.
The inverse of the latter is now composed of 1/T" B , related to perturbations
of all bodies other than m in the distribution mT q up to the maximum mass
mi~ 10" 3 m (see Chapter 8), and 1/7T>, related to perturbations of the embryo
m. From (71) we have, in place of (72),
r D =
d~ q
i>3
2 — g 8it/iG2 mi p" »
(76)
85
where p" is the density due to all bodies other than m.
To determine T* D it is necessary to evaluate the frequency of encounters
between the body and the planet embryo at various distances from it. We
will assume that the bodies move completely at random over the entire
planetary zone, which we will treat as closed. This is facilitated both by
mutual perturbations among the bodies and by perturbations emanating from
the planet embryo. Opik (1951) mentions, for instance, that under the
influence of the Earth's secular perturbations the line along which the
inclined orbit of the body intersects the plane of the ecliptic (nodal line)
describes a complete turn only once in 6 . 10 years. Since m' <w, when
evaluating the frequency of collision of the bodies m' with the planet embryo
they can be treated as point masses. If a point is traveling in space with a
velocity v and encounters on the average n orthogonally placed areas s per
unit volume along its path, the mathematical expectation for the point enter-
ing this small area in the time t will be given by nsvt. We assume that in the
zone adjacent to the planet, which for a body moving with velocity v has a
volume
o 4 '
there exists a single planetary embryo. Setting n~\jSH and s^=2nDdD, we
obtain the mathematical expectation of the number of passages of the body at
an impact distance between D and D + dD from the planet, given by
ro(= y, (77)
Here o is the total surface density of matter in the zone, including the planet
m. As the body approaches the planet its relative velocity vector v rotates
through the angle W:
The mathematical expectation for deflection ^ within the time t for numerous
encounters at various impact distances D is given by the expression
where
f — [ Xdx — 1n 1+Jjf * , 1 , 01 D M
(79)
The relaxation time T* D is given by the condition
2rin»V = 1. (80)
86
Consequently
r * QPv* ( p i \
If we take the collision cross-section ttr 2 (l +20") in (77) instead of InDdD ,
we obtain the mathematical expectation of the number of collisions between
the body and the planet embryo within the time t. Setting it equal to unity,
we obtain
-•-- QP - (82)
4Ttoor2(l-f 2ft*)
The time x* within which a body m! colliding with smaller bodies will acquire
mass m! can be determined from (73):
\mj ** Pm' 9\mJ V Qj'
(83)
Let us set
n = n/ Z . (84)
Then
*- 1 /(TT + T7)- rw(1 + * (84 ' }
As for (74), from (76) and (83) we obtain
Consequently,
e ,/2 /^i V»_ Q2 _ 3-ff bU (m,\i-*U .
where
The additional factor (rr^jmf* gives us a substantially larger value of 9".
The velocities of the bodies turn out to be nearly the same as if there were
no embryo m, and the main body governing the velocities, according to (7. 12),
should be the body m x . Therefore if we set v=t\lGmJfcr lt expression (86) will
yield values of 6 X comparable with the values of obtained earlier.
The relative velocity of a body increases because it draws closer to other
bodies at various points along its elliptical orbit (various q>, R, v). At the
initial stage these conditions of encounter are fulfilled automatically, since
the bodies are sufficiently numerous inside the zone. At the final stage they
will obviously be fulfilled as long as T" D <^T%, i. e -> as lon g as m is sufficient-
ly small. The relative velocities are then given by (86).
87
The encounters of the bodies with the planet embryo m traveling along a
nearly circular orbit take place in practice at a fixed distance from the Sun
and, therefore, for a fixed value of the relative velocity v. Encounter of a
body with the embryo will be followed by a change in v only if its orbit is
altered by other bodies. Therefore if there is only one large embryo m
inside the planetary zone and for this embryo 7i<rj, the effective relaxa-
tion time Tl, in encounters between the bodies and embryo will be equal to
the relaxation time Tl. Consequently in this case, in expression (86) one
should take x = 1 for sufficiently large m.
There is reason to believe, however (see Section 26), that the planetary
zone originally contained several embryos. As the embryo masses grew
their source zones aggregated, the number of embryos decreased, and finally
the largest of these became the planet. The intervals R i+1 —Bi between
adjoining embryos, amounting to several times the radius r L of the largest
closed Hill region, were smaller than the width 2A/? of the region within
which the bodies moved (see (9.9)), and therefore 2 to 3 embryos could have
coexisted inside it simultaneously. To evaluate x g under these conditions we
will assume that, in addition to the "main" embryo m , the planetary zone
contained n x embryos of mass mln 2 . The relaxation time T* m associated with
the action of these additional embryos on the body is given by
T* ~ "1 T* ( 87 )
ni ( 1+ -rr)
For the case TKT m B1 <Ti one can take -: ff ^r m /2. Then instead of (86) we
obtain
5$if£_iy^iY" ,/ ' (88)
The number n± of embryos can be estimated if we assume that their mass
amounted to the fraction a of the total mass Q— m of material not contained in
m. Then
H _ .tg-»)- t (89)
and from (88) and (75) we obtain
ft" 2 ~ 5*/* (^Y* 1 * "2 _ 10(2 — g) n 2 / 4 /«i \f-V, fl , a (90)
~ 2/l ^ » A } 21nnj, \ <3-d«(l+21nn 1 // 3 )/ s \m )
The values of 6" turn out to be several times larger than the corresponding
values of 0' according to (75). For n, = 5, a = 0.5 and m l = iO' Z m, the value
of 6" is approximately 4 and 2 times larger than 8' if p= 3.5, respectively.
For r ~ 3 ■ 10 cm the values of 6" lie between 3 and 7 for p= 3 and between
4 and 8 for p = 3.5.
In conclusion we note that the foregoing estimates of were based
exclusively on the increase in relative velocities of bodies within the frame-
work of a two- body problem, where encounters are characterized by rotation
88
of the relative velocity vector without change in its absolute value. No
account is taken here of the role of multiple encounters of a body with the
planet, which can involve systematic changes in the orbital parameters.
There is a known tendency, for example, for planets and satellites to "enter
into resonance," leading to the establishment of commensurable periods of
revolution among adjacent bodies. The smaller body will experience "inducec
eccentricity" of orbit, even when the orbit of the larger body is strictly
circular (Goldreich, 1965b).
It is therefore not excluded that the actual values of 6 were somewhat
smaller than the values obtained above.
89
Chapter 8
STUDY OF THE PROCESS OF ACCUMULATION OF
PROTOPLANETARY BODIES BY THE METHODS
OF COAGULATION THEORY
24. Solution of the coagulation equation for a coagulation
coefficient proportional to the sum of the masses of the
colliding bodies
The size distribution function of bodies is one of the most important
characteristics of the protoplanetary cluster. On it depended, to a large
extent, the relative velocities of the bodies, the extent of their fragmentation
in collisions, the rate of growth of planetary embryos, the transparency of
the cluster, and the formation of satellite clusters. The geophysical conse-
quences of the accumulation of the Earth also depended largely on the sizes
of the bodies that formed the Earth. This applies first and foremost to the
initial temperature of the Earth and to the primordial inhomogeneities of its
mantle. Studying the size distribution function for bodies in the process of
planetary formation is therefore to be regarded as one of the primary tasks
of planetary cosmogony.
The process of aggregation of protoplanetary bodies is similar in some
respects to the process of coagulation studied in colloidal chemistry, and
also to the process of growth of rain droplets studied in meteorology. Thus
it is natural to adopt the methods of coagulation theory for its investigation.
Unfortunately, in view of the uniqueness of the accumulation process, no
single concrete solution of a problem in coagulation theory can be used to
describe it. One can only exploit the most general relations of the theory
(i. e., essentially, the method) to construct concrete equations and attempt
to solve them.
Coagulation theory is concerned for the most part merely with the fusion
of particles. In the accumulation process, on the other hand, disintegration
(fragmentation) of colliding particles is also important. Accounting for frag-
mentation adds very considerably to the complexity of the investigation. It is
therefore expedient to begin with the simpler instance of accumulation of
bodies that have not experienced fragmentation.
In coagulation theory, the foundations of which were laid by Smoluchowski
(1936), the equation of chemical kinetics is usually written in the "discrete"
form (see, for example, Chandrasekhar, 1943)
5979
90
where v 4 is the number of particles in an element of volume of dimension k,
i. e., consisting of k elementary initial particles, and A tJ can be termed the
coagulation coefficient. This system of equations was solved by Smoluchow-
ski for the simplest case of A kJ =A Q = const for a monodisperse initial state.
There also exists an integral form of the coagulation equation (see, for
instance, Schumann, 1940; Todes, 1949):
a»(m t <> l l A < m > y m _- TO ')„( m ', t)n(m — m' t t)dm F —
at & J
o
— n (ntyt) \ A (m, to') n (m\ t)dm!\ ( 2 )
o
unlike v 4 , n (to, *) is a continuous function of the particle mass m representing
the number of particles of mass to (more precisely, within the unit mass
interval Aro = l) in one cm 3 ; A (to, to') is the collision and aggregation probabil-
ity for particles to and m' (coagulation coefficient). The first term on the
right in (1) and (2) represents the number of particles of mass to (with the
index k) formed per cm per sec as a result of the aggregation of particles
of mass m' and to— to' ( t and j=k—i). The second term represents the number
of particles of mass m combining per cm per sec with other particles,
acquiring a different mass as a result.
A different type of equation has been used in astronomy to study the
growth of interstellar particles (Oort and van de Hulst, 1946). To facilitate
comparison with (2), we will convert it from an equation for n(r t t) into an
equation for n(m, t). It then becomes, in the absence of fragmentations,
dn (m, t) d_
dt dm
n (m, t) \ A (to, m!) m'n (m ! y t) dm 1 —
o J
— n (m, t)\A (m, m!) n (m', t) dm 1 .
The integral in the first term on the right is equal to dmldt, and the first
term as a whole represents the changes in n(m, t) which stem from the fact
that the masses m of all bodies increase due to absorption of all bodies
smaller than m during collisions. More briefly, without the last term
equation (3) becomes a one-dimensional continuity equation in which n(m, t)
represents density and dmldt the analog of velocity. The second term on the
right represents the variation in the number of bodies of mass m due to the
fact that such bodies fall on larger bodies. An equation such as (3) is also
used by Piotrowsky (1953) and Dohnanyi (1967) to study the process of aster-
oid disintegration. In meteorology Telford (1955) solved a similar equation
without the last term on the right.
The fundamental difference between equations (2) and (3) is that, accord-
ing to the former, different bodies of identical mass m will have different
fates depending on what other bodies they collide with. This results in a
rapid increase in the size dispersion of bodies in the system, as well as in
more rapid growth of the few bodies that accidentally experience more
frequent collisions. By contrast, according to equation (3) all bodies of
mass m will grow in the same "mean" fashion due to the settling of smaller
bodies. This is an accurate description of the growth of the basic mass of
91
bodies, but certain important laws of random stochastic processes are over-
looked in such averaging. Equation (2) is therefore to be preferred.
The assumption that A (m, m') is constant is entirely inapplicable to our
case. For A (m y m')^ const the problem becomes very complex and equation
(2) is usually solved by approximate and numerical methods (e.g., Pshenai-
Severin (1954), Das (1955)). But such methods can describe only the early
stages of the process and do not permit us to follow the growth of bodies over
vast time spans, corresponding in our case to a mass increase by 8— 10 orders
of magnitude. It is therefore desirable to obtain an exact analytic solution,
even if it means taking only a qualitatively reasonable expression for the
coagulation coefficient.
When allowance is made for the gravitation of the bodies, the coagulation
coefficient can be written as
A (m, m>) = w (m, «') n (r + r') 2 [l + ^ + ^' j V, (4)
where w (m, m') is the probability that colliding bodies will combine, r and r'
are the radii of bodies of mass m and m' respectively, and V is the velocity
of the body m relative to bodies m' before encounter. In the absence of frag-
mentation one can take w(m, m')ml. The relative velocities of the bodies will
depend on their masses, though not strongly (see the chapter on velocity
dispersion). Quantity A (m, m') depends mainly on the mass and radius of the
bodies, which vary over a wide range. For small bodies the collision fre-
quency and A(m, m') are determined by their geometrical cross-section
n{r+r'Y and are approximately proportional to mV For large bodies the
cross-section increases due to gravitation (the second term in square
brackets predominating) and A(m, m') is approximately proportional to
(m+m') (r+r') } i.e., ~-m*t> for m' <^t m.
The author (Safronov, 1962a) and Golovin (1963) have obtained an analytic
solution of (2) for a coagulation coefficient proportional to the sum of the
masses of the colliding bodies,
A(m, m')^ («+*'). (5)
where A x ~ const. This expression for A (m, m') is a kind of "average" between
the above expressions for small and large bodies. It gives a qualitatively
accurate general pattern of the mass dependence of A(m t m'). For this value
of A (m, m') equation (2) becomes
m 00
^TdP ^tI"*™'' 0»( w - m ' t)dm'— n{m t t)\(m + m')n(m' t t)dm!. (6)
This equation can be solved with the help of the Laplace integral transform.
We begin by dropping the second term on the right. Integrating the left and
right hand sides of (6) with respect to m, we obtain an expression for the total
number of bodies in the system:
00 m 00 00
^j$-= \ -j- dm \ n (m\ t) n (m - m\ t) dm' - J n (m) dm J (m + m!) n (m\ dm'.
92
The second integral on the right is equal to2/Vp. Changing the order of
integration, the first integral becomes
Y \ n ("*', t\ dm! \ n (m — m\ t) mdm =
00 00
~y\ n(m',t)dm' jrc(m, t)(m-\-m')dm— /Vp.
Therefore
where
d ^l = -_Ar p and TV = N #-'#', ( 7 )
^o = ] ra ( m » 0) dm, p—\mn (m, *) dm = const. ( 8 )
o o
We replace the required distribution function n{m, t) by the new variable
g(m t t) defined by
n(m, 0=e ym/MlP '£(m, 0- (9)
Equation (6) then becomes
^*teJL=- ][,(„_*., tup, t)d m >. do)
We now replace the time i by a new independent variable x :
d* = 9AiTs-dt, x = l — «-^* = l_ *. (H)
When t varies from to oo, the variable x varies from to 1. Instead of
(10) we obtain
T^-^\e(m-m!,,)g(m',^dnJ, (12)
where the same symbol g{m> x) denotes the new function which emerges when
t is replaced by x in g(m, t) with the aid of expression (11). The Laplace
transform is applicable to equation (12), From the inverse function g(m, x)
we pass to its representation G(p, x):
00
G(p, x)= [e-r-gim, x)dm. (13)
93
Multiplying (12) by e~ pm and integrating with respect to m, we find
00 m
p ^ = y \ e~ pm mdm J g (m\ *)g{m — m\ x) dm! —
oo oo
= T$ e ~ pm 'g( m '> x ) dmf \ e-P( m -^g(m — m\ x)mdm =
m'
oo oo
= ~ J e-P m 'g (m' f t) dm! J (m + m!) e~^g (m, %) dm = -G ^ .
For the representation G(p, x) we thus obtain a quasilinear partial differential
equation
p|2 + 6g = 0. (14)
The general solution of this equation has the form
G{p, t) = G [ P p-G(p, x)x], (15)
where the arbitrary function G (x) is determined from the initial data.
According to (13), (9) and (15),
G{p, 0)=Je-^(ro, 0)dm = \ «"^^"n(m, 0) dm = G (pp). ( 16 )
From (15) and (16),
G(p, t)= J r IP ^ (f ' T)+ ^Fn(m, 0)dm. (17)
From here G (p t t) can be determined explicitly only in a few instances.
If, for example, we take the initial distribution to be
n(m, 0) = am-'<r 4m , (18)
G(p, t) will then be given by the expression
co». ^{p+^-^j^y =ar ( i_,), (is)
where g <1. For q= 1 /2 and g =-1 one obtains a cubic equation for G (p, t)
which can be solved. For other values of q one obtains algebraic equations
of higher order for G (p, r). We will confine ourselves to the simplest case
where q = 0. Then
n(m t 0) = ae-*", (20)
a = yVJ/p = ^ /m , b = NJ ? = -L, (21)
where m is the mean body mass at the initial instant; Introducing n(m f Q)
94
from (20) into (17) and integrating, we obtain
G (P' T ) == p+2t- P -.,G(p, t) « (22)
whence
G(P, ^) = -^[p + 26±^(p + 26)«-46»c]. (23)
A characteristic property of the quasilinear equation (9) is the indeter-
minacy of its solutions (I. G. Petrovskii, 1953). This is evident in the
solution (23) from the two signs preceding the square root. The branch
point occurs at p = —2b (l — \Jx) . It shifts from p=— 26 for t = to p = for
t=1, i. e., for t—oo. A definite single-valued solution can be obtained only
in the case where the branch point constantly lies beyond the region of values
of p considered here. On the other hand, the function G (p, t) obtained by
means of the Laplace transform is defined in the complex half- plane Rep >
> s , where s is the growth exponent of the inverse function g (m, t) . From
solution (26) obtained below for g (m, t) it is seen that s Q =— 26 when t = and
s = when t — 1. Thus the two restrictions on p coincide at the ends of the
interval of variation of t. By taking p > 0, we can simultaneously satisfy,
for all values of t, both the condition imposed on the inverse transform and
the condition that there be no branch points in the domain of the variable
under consideration. The latter is made possible only by the fact that the
quantity x in equation (14) varies in a bounded interval: as the time increases
indefinitely, t -* 1. This may seem to be accidental from a formally mathe-
matical standpoint. But from the standpoint of physics this result is natural.
The equation in question describes a definite physical process, and indeter-
minacy of the solution would indicate instability and the presence of special
points in the process.
We note that a similar result will be obtained for other initial distributions
n (to, 0). It is easy to show, for example, that if n (to, 0) is of the form (18) or
a 6-function, then for real p, points on the envelope of characteristics will
lie within the region of negative p for all t< 1. In the case of complex p, for
sufficiently large Re p the real and imaginary parts of G (p, t), as is seen
from (17), will be small on the correctly chosen branch. But when the
imaginary part is small the solution for G (p, t) differs only slightly from the
solution for real p.
Consequently, even for complex p the lower boundary of Re p can be
chosen so that the branch points of G(p, x) always lie outside the region
under consideration.
Of the two branches of solution (23), only one is suitable (the one with the
negative sign in front of the square root), since only then will G (p, x) remain
bounded for t — and tend to zero for p -> oo, as should be the case according
to (13). Consequently, the required solution of equation (14) for the initial
distribution (2 0) should be of the form
G(p, t) = £0 + 26-V(p + 26)«-4W]. (24)
To pass from the representation G (p, t) to the inverse function, one must
carry out an inverse Laplace transformation. Let us use the transformation
95
given in the manual of Ditkin and Kuznetsov (1951),
p_tf^r=?^.L/ i(am)i (25)
where /i(x) is a modified Bessel function. Using the displacement theorem,
we obtain the following expression for g(m, t):
g ( m , x) = JZ±= e- 2 *"*/ 1 (2bm >/i~). (26)
m vx
Finally, with (9), (11) and (21) we pass to the required distribution function
n (m, t) = Nq (1 ~Z t) e-t W*»»/ (2frm \/7). (27)
m Vt
The function 7, (#) has the following expansions:
for x <€ 1
and for x^> 1
/ to— -EJl i < J / 2 > 2 i -(*/ 2 > 4 i 1
'■w^l 1 -™-) < 29 >
Correspondingly, the mass distribution function of the bodies (27) can be
approximated as follows:
for 2m^<m„=l/6
and for 2/nV^">m
n{m t z)&N Q b(l—z)e-W> m , (30)
„(«, x)^ ff »< 1 -;) m-V (l -^ )2t ". (31)
2 v^Tt '«
It is only at the early stage of the process of aggregation and for small
values of m that the condition that 2mfi <^m is met and that the distribution
function is exponential. For most of the region of values of m and t , one
can use expression (31), which is the product of a power of m by an exponen-
tial function. For large t the value of x is close to 1 and (l-y^r - ) 2 is very
small. In this case, therefore, the exponential function will begin to play
an important role only for the largest m. The condition n(m)am J/ ' is met
over nearly all the interval of variation of m (except for the largest and
smallest m). We note that this power function is close to the mass distribu-
tion obtained observationally for small bodies in the solar system — comets,
asteroids, meteorites reaching the Earth. The exponent -72 of m is indepen-
dent of the parameters a and b of the initial distribution and is apparently
determined only by the form of the coagulation coefficient A (m, mf).
In distribution (31), a considerable portion of the mass of the system
consists of large bodies; with time, moreover, the relative mass of the
96
FIGURE 4. Mass distribution function fi (m)Am=mn (m) m y
for bodies, obtained from the analytic solution of the
equation for coagulation without fragmentation. Quan-
tity m l is the mass of the largest body in the distribution
and is taken as unity along the horizontal axis:
1 - N - N (t = 0); 2 - N - NT' N ; 3 -
N = 10-' N ;
large bodies increases. Figure 4 indicates the mass distribution \i (m)=mn (m)
at different instants corresponding to reduction by factors of 10, 10 2 , 10 3
and 10 5 in the total number TV of bodies in the system compared with the
initial number N . The unit of measurement of m is taken to be the mass
of the "largest body" m lt obtained by integrating the "tail" of the mass
distribution function, which contains one body:
CO
m x •=■ \ mn (m) dm,
where
J n(m)dm = i.
*.
It would of course be rash to apply this result directly to the process of
planetary accumulation, as no account has been taken here of the fragmenta-
tion of colliding bodies, which increases the amount of fine substance in the
system. But the result that large bodies played a considerable part in the
accumulation process seems to be correct.
25. Asymptotic power solutions of
the coagulation equation
Owing to the complexity of the coagulation equation, especially when
fragmentation is allowed for, there is very little hope that an analytic solution
for it will be found. Hence the importance of qualitative methods of investi-
gating the equation that shed light on the nature of the distribution function
without seeking an actual solution. Such methods include attempts to seek
partial solutions that could be regarded as asymptotic. Piotrowsky (1953), for
instance, having proposed a power form of solution to his equation for the
size distribution of asteroids involving relatively simple analysis, arrived
at the conclusion that the radius distribution of asteroids must tend to a
97
power distribution with exponent p= 3. Our equation is considerably more
complicated but it too can be submitted to analysis of this kind.
a) Accumulation in the absence of fragmentation. Consider equation (6).
For greater generality we will take finite limits of integration:
dn J m dt t] = ! y\ w K. t)n(m — m\ t)dm< — n (m, t)\ {m + m')n(m , t t)dm ! ,
(32)
where m and M are the lower and upper bounds in the distribution. We
assume that at a certain instant / the distribution function has the form
n (m) — cm ? ,
(33)
where c and g are independent of m. Introducing the value of n (m) in (32)
and setting m'lm — x, we obtain
j^ = m* \ *"' (1 - xy dx-\(l+x) x-<dx
_ w m
^-•[(i-«r-ij«te+ r ^ 7 [(^)'- , _2^]
m
(34)
Expressing c in terms of the total mass of material per cm , which we will
regai'd as constant, we obtain, for q ^ 2,
2~q
and
A } ndt
1— (m,
-g)P (H\ 2 - q F(a -^ ^
(34'
Relation (34) tells us in what direction the variation of n (m) proceeds. The
second term on the right, which is independent of m, leads to the same
relative reduction in n (m) for any m, i. e., it characterizes the decrease of
c in (33). The first term gives the relative variation in n (m) depending on
m. It therefore produces variation in ^q . If, for instance, it is positive and
increases with increasing m, the fraction of large bodies will increase and
q will decrease.
In the general case the variation in q is different for different m y the mass
distribution of the bodies deviates from the power law immediately, and
relations (34) and (34') become inapplicable. But for the initial instant when
the distribution by assumption follows a power law, these relations hold true
and indicate the direction of variation in q. By introducing various values of
q in them one can try to obtain an asymptotic distribution.
98
In order for a power solution of the coagulation equation with an exponent
q to exist, it is necessary that for this value of g the right-hand side of (34)
and (34 ! ) be independent of m. Of greatest interest, therefore, are the
values q = g that are roots of the equation F(q, -^-, ^j= 0. If the coagulation
equation is of a form such that F is independent of m s then the root q will
also be independent of m. Then the power distribution (33) with exponent q
is a solution of the coagulation equation. This holds, for example, for the
coagulation equation under consideration when m = and M=co. For F (q) to
converge in this case, it is necessary that q < 2 (but cF(q) in (34') will con-
verge even for q > 2). Then
Since the solution to equation (6) corresponding to the initial distribution (2 0)
is already known (see (27) and (31)), our first step must be to check whether
the value q = % is a root of equation (35). A simple substitution will convince
us that this is so. Consequently, the function n(m) — cq- 3{ * is truly a solution
of (6).
However, not every root q of F (q)= will give us an asymptotic solution.
An asymptotic distribution should be stable, i. e., distributions with q
approaching q should tend to it. For q < 2 this condition will be met if
F' (9o) > 0. Then for q < q Q we will have F (q)< and, from (34'), the relative
decrease in n will increase with m; therefore q should increase until it
reaches q . For q > q we have F (q)> and dn/ndt increases with increasing
m. The relative fraction of large bodies increases and q decreases until it
reaches q . Consequently, q ~* q Q from both sides and the distribution cm 10
is indeed asymptotic. By contrast, for F' {q Q )< the exponent q gives an
unstable solution: for q <q Q , q decreases while for q> q , q increases.
Such a solution could not be asymptotic since solutions close to it would
diverge from it in the course of time.
For q 9 >2 the solution is asymptotic if F ! (q )< 0. Lastly, q = 2 is a special
value. It gives an asymptotic solution for F (q=2) < 0.
Thus the condition that an asymptotic solution of the power form type
should satisfy can be written as:
F(q) = 0, F'(q)>0
for
?<2,
F(q)<0
for
? = 2,
F(g) = 0, *»(?)<
for
?>2.
(36)
From (35) it is seen that when q decreases from % to 1, F(q) decreases
from to -co. Thus F'( 3 / 2 )> 0. This means that q = z /2 should give an asymp-
totic solution.
This emerges from the behavior of solution (31). From (11)
(A /~\2 u 1 / tf \2 m i N m
In the course of time m increases, but then m also increases, while N/N
decreases. The role of the factor with a power of m in (31) will therefore
decrease constantly for the same m/m . The body distribution will tend to
follow a power law over an increasingly large interval of values of m.
99
The foregoing discussion is largely formal, since the power distribution
law is physically inapplicable over the entire infinite interval of variation of
771. The assumption m = and Af=oo means that either p=oo or c= 0. In the
former case the second term F 1 in (34) will diverge, while in the latter for
q < 2 the first term in (34') will tend to zero. Nevertheless the above method
of qualitative analysis of the coagulation equation makes it possible to obtain
asymptotic solutions of power form. While the power solution is in itself
physically inapplicable, it is the limit of real solutions that do not possess
its defects. Thus in solution (31) the principal term is the power mr*** and
there is an additional exponential factor which removes the divergence of the
asymptotic power solution. The form of the additional factor is probably
determined by the form of the initial distribution n(m, 0), whereas the charac-
ter of the asymptotic solution reflects the properties of the equation. In the
presence of the exponential factor the difference between distribution func-
tions with finite and infinite limits M is insubstantial, since n (m) decreases
rapidly and not a single body remains throughout the interval of values of m
greater than a certain M: ^n(m)dm< 1. In practice a distribution with M=co
M
is equivalent to a distribution with finite M to which has been added, in the
region of largest m~M , an additional finite mass given by ^mn(m)dm. But
M
mathematically the difference between these distributions is considerable:
for the one there exists an asymptotic power solution, for the other no such
solution exists and all solutions are more complex. Since, however, for
m^O and M^ozthe function F(q,~ t 2fl) i s not very different from F (q) in (35),
if m <4 m < M one might expect that an asymptotic solution for this range of
values of m should be close to the asymptotic solution for m Q = and Af=oo,
i. e., close to a power function with q Q = 3 /2.
b) Allowance for the fragmentation of colliding bodies. Fragmentation of
colliding bodies played an important role in the process of planetary accumu-
lation. By increasing the amount of fine substance in the system, fragmenta-
tions exercised considerable influence on the size distribution function of the
bodies. Quantitative treatment of this effect is very difficult. Even without
allowing for fragmentation, the coagulation equation is too complex for there
to be any hope of obtaining an analytic solution. Moreover, the fragmentation
process itself has hardly been studied, and no reliable data exist concerning
the size distribution of fragments in impacts from large bodies. A well-
known logarithmic law of size distribution was obtained by Kolmogorov from
purely probabilistic considerations for the case of multiple fragmentations of
particles experiencing any kind of disintegration. A more general expression
was later found by Filippov (1961).
We note that the probability for disintegration of colliding bodies with a
significant gravitational attraction varies for different mass ratios. It is
known that a particle striking the surface of a larger body at a velocity of
5— 10 km/sec will form a crater, scooping out from it a mass 2—3 orders of
magnitude greater than the mass of the particle. Therefore the mean veloc-
ity of ejection will be 1 — 1.5 orders of magnitude less than the velocity of the
100
impacting particle. If the body is massive enough the ejected matter will be
unable to overcome its attraction and will fall back on the body. In this case
fragmentation in the above sense (i. e., of disintegration) will not take place.
A different picture emerges for the collision of bodies of comparable mass
at the same velocity. Here the impact energy per unit mass for both bodies
is much greater, and correspondingly the velocity of dispersion of matter
will be considerably larger. Consequently, the probability for disintegration
(fragmentation) in collisions between bodies of comparable mass is consider-
ably larger than for substantially different masses.
Let w (to, to') be the probability that the bodies m and to' will combine in
collision and 1— w (to, to') the probability that they will fragment. Further,
let n^m, to") be the distribution function for the mass to of the fragments
resulting from collisions between two bodies of total mass to". Obviously,
to < to" and
j n x (to, to") mdm = to". (37)
o
Allowance for fragmentation introduces the following change in the coagu-
lation equation (2). In the first integral the integrand is multiplied by
w (to\ to— to'). The second integral remains the same. A third term is intro-
duced, characterizing the increase in the number of bodies per unit time due
to fragmentations. Two- body collisions with total mass to" gives an incre-
ment n x (to, to") N (to") in the number of bodies to, where N (to") is equal to the
first integral of equation (2), in which the integrand has been multiplied by
1- w (to', to"— to') and to" replaces to. The total increment in the number of
bodies to is obtained by integrating this expression over all to" >to. Conse-
quently, the equation has the form
dn{ ^; t] = [ w(m\m—m!)A(m!,m-m!)n(m!,t)n{m~m\t)dm!—
at J
o
CO
— n(m, t) \A(m, to') re (to', t)dm! +
o
oo m"12
+ j n x (to, to") \ [\—w (to ; , to" — to')] A (to', to" — to') X
X n (to', t)n (to" — to', t) dmfdmf. (38)
A preliminary examination of this equation was carried out by the author
under the assumption that
w{m', to — to') = 1 for to'<-^-(1 — a)
and
w(m', to-to') = for m '>^(l-a) (39)
(total disintegration of bodies of comparable size). The distribution function
over to of the fragmented material was taken as
nj(m, to") = cmV*» M /"', (40)
101
and the coagulation coefficient in the form (5). The body mass n (m, t) =
= mn x (m, t) was taken as the unknown function. After transformation to
remove the divergence of the first two integrals for m -* 0, the coagulation
equation assumed the form
A-iOt J m
CD
/ \ f f 1 ( m ') dm ' i ,v
J m '
-\-b 2 f <?-*.«/«" (
Mm', <)Hm-m', 'Wrf m ».
(41)
Searching for an asymptotic power solution indicated that condition (36) is
met only for very small values of a. The root of the equation F (q, a. . .)=
which yields an asymptotic solution moves with increasing a from g = 3 /2 for
a= toward smaller values of q. For a > l(f 2 the function F {q, a) has no roots
in the region of values of q<2. To check whether this resul+ could have been
due to the form of the distribution function adopted for the fragmented bodies
(40), a similar calculation was carried out for the following distribution:
M«.m'0 = c(-£)"*.
(42)
It was found that, in this case as well, condition (36) is met only for small a,
and the roots q move with increasing a toward smaller q. Table 10 gives
the values of q for ^(m, m") = c(^X qi as a function of the parameters a, q Y ,
and M ! = Mim.
TABLE 10
M f
OC
Qi
o.ot
0.05
0.07
0.10
0.12
0.15
0.4
0.4
0.7
1.2
1.5
00
100
00
100
100
1.5
1.5
1.47
1.45
1 43
1.48
1.44
1.46
1.47
1.40
1.45
1.37
1.42
-
For AT = oo a solution exists only for q x < 1, while for M'^co there is no
exact power solution, since q is slightly different for different AT. Unfor-
tunately, solutions that hold only for small a cannot describe gross proper-
ties of the system associated with fragmentations.
102
We were able to obtain an asymptotic solution for larger values of the
fragmentation coefficient a for a power function n x (m, m") of the form (42),
for the special case where q l =q (Figure 5). Below we list the values of the
exponent q 01 of this power solution as a function of a :
a 0.1
A = A 1 (m + m') 15 1.55
A^A^m. . . . - 0.94
0.2
1.62
1.13
0.4
1.74
1.25
0.6 08
1.85 1.94
1.34 1.41
as a* as as /Ja
FIGURE 5. Dependence of the exponent g in the
asymptotic power solution of the coagulation
equation (with fragmentations) on the parameter
a for the case where ft equals the exponent of
the required distribution.
In the third row of the table we list
for comparison the values of q n f° r * ne
same premises but a slightly different
coagulation coefficient: A=A 1 m, where
m is the mass of the larger of two
colliding bodies. The differences in q 01
are easy to understand: as the bodies
being fragmented are of nearly the same
size, for Acc(m+m') the fragmentations
are nearly twice as intensive as for
Accm , and q 01 should be larger. The
form Ai-im+m') is closer to reality.
The most likely values of the exponent
of the asymptotic power solution are
apparently g 01 = 1.8—1.9. The exponent ?<>i remains less than 2 despite the fact
that ft must necessarily increase with a. The more general case of q^=q x has
been analyzed by E. V. Zvyagina. She found that q tends to a certain value
between q 01 and q x which is closer to q x as m decreases. Since the values of
q depend on m, there is no exact asymptotic power solution. Still, from the
foregoing results one can conjecture that approximation of the size distribu-
tion function by a power function is permissible in a system of bodies with
fragmentations and that the exponent q of this function will lie between 1.5 and
2 (more likely closer to 2) depending on the intensity and character of the
fragmentations. In the large body region (m^M) the deviations of the distri-
bution functions from the power function will be maximum, but in the region
of small bodies ( m < M) the power approximation is entirely satisfactory.
Roughly the same values for q are obtained in the limiting case where
there is no accumulation, only fragmentations taking place. An equation of
type (3) was studied for asteroid fragmentation by Dohnanyi (1967) as well as
by Piotrowsky. However, Dohnanyi assumed that fragmentation produced
not only fine particles but also larger fragments with a power mass distribu-
tion law of exponent g^l.8, as follows from experimental data. For the
stationary case dn/dt= 0, Dohnanyi obtained a power solution with exponent
It should be mentioned that the foregoing results regarding q agree with
factual data on distributions of asteroids (Piotrowsky, 1954; Jashek, 1960),
meteorites (Braun, 1960) and comets (Opik, 1960), which lead to $« 1.6—1.8.
An attempt was made by the author to calculate the function n (m, t) on a
BESM-2 computer for the above premises regarding A, w and n lt i. e., in
the form (5), (39) and (40). Simplifications made in the program did not allow
the variation of n (m, t) to be followed over a large time span. Nevertheless,
103
-to -
FIGURE 6. Time variation of the mass distribution function for bodies for two nearby values
of the parameter a characterizing the fragmentation intensity of colliding bodies (calculations
performed on a BESM-2 high -speed computer;.
it was found that the variation of the distribution function was substantially
different for different values of the fragmentation parameter a. For a < 0.78
accumulation predominates and in the course of time the number of larger
bodies increases, the entire distribution shifting toward larger m. For
a > 0,78 the picture is reversed: the number of large bodies decreases and
the distribution shifts toward small m. The value a^0.78 corresponds to
the root q a characterizing an unstable solution, according to the terminology
introduced earlier (Figure 6).
Thus from preliminary analysis it is already apparent that fragmentation
of colliding bodies has a substantial influence on the size distribution func-
tion established during the accumulation process. It is important that
information on the general character of this function should be found, and
therefore data regarding the nature of fragmentations in collisions must be
made more precise. It is necessary to determine to what extent the avail-
able laboratory data on fragmentations of small particles are applicable to
the description of fragmentations of large bodies. In view of the great
difficulties that arise in the solution of the coagulation equation, it becomes
especially important to develop methods for analyzing it qualitatively and
search for asymptotic (and in particular, power) solutions.
104
Chapter 9
ACCUMULATION OF PLANETS OF THE EARTH GROUP
26. Growth features of the largest bodies
When the distribution function for protoplanetary bodies is studied by the
coagulation theory method, certain fundamental laws of the accumulation
process fail to emerge. Asymptotic solutions are a good representation of
the distribution function in the region of small and medium- sized bodies.
As to the growth of the largest bodies, much remains obscure. The concept
of a distribution function is applicable only as long as the number of bodies
(within a given size interval) is large enough to permit the use of statistics.
But statistics cannot be applied to the largest individual bodies. Yet these
may be the most interesting of all, since one of them eventually becomes
the "embryo" of the future planet. It is therefore fitting to dwell at greater
length on their growth pattern.
It is widely held that if two bodies of different size are placed in a medium
which supplies them with material, then the masses of the bodies will tend to
equalize as they grow. This is valid as long as the rate of mass growth of a
body is proportional to its geometric cross section, i. e., to the surface
area of the body. For the smaller body the surface area per unit mass is
greater, and its relative increment should be greater, than for the larger
body. However, these considerations do not apply to the largest bodies.
Owing to gravitation their effective cross sections are considerably larger
than the geometric cross sections and are proportional to a higher power of
mass than the first. In this case the mass difference (ratio of larger to
smaller) will increase and not decrease with time.
If a body (m, r) is large compared with other bodies m! , then m+m't&m,
r j rr '^ r) an d its collision cross section, after allowing for gravitation, can
be written as
«p««-(i+^) 1 (i)
where V is the velocity of incident bodies relative to m before approach to m.
The velocities of very large bodies are usually small, and V is, in practice,
the mean velocity v of incident bodies relative to the circular velocity.
Expressing it according to (7.12) in terms of the mass and radius of the
largest body in the form v 2 =Gm/Qr, we obtain
J 2 «r 2 (l+26). (2)
105
Since 6 is of the order of a few units, the second term dominates and
For the body next in size (m^) situated in the same zone, we obtain
Z ? «r ? (l+^i) = r?(l + 2e-i). (4)
As long as m x is comparable with m, the second term on the right in (4) will
predominate and therefore
The ratio of the effective cross sections of the largest bodies is proportional
to the fourth power of the ratio of their radii. Therefore
dm i m ^ r ^ | (5)
It is easy to show that the largest body m will then grow more rapidly
than the body m 1 both absolutely and relatively, i. e., the ratio mlm 1 increases
with time.
Consequently, the largest body outstrips the others as it grows, pulling
apart from them, as it were. However, the difference between the masses
m and m l can increase only to a certain limit. When m l becomes much
smaller than m, the second term in (4) becomes smaller than the first and
the collision cross section of m 1 approaches the geometric cross section:
l\~r\. (4')
Then l 2 jl\^2§r 2 jr\ and
J^L^ 26 i (6)
The ratio mlm x stops increasing when the right side of (6) decreases to unity,
i. e., when
r^26r 1( mm(2Qfm v (7)
The values 0^3—5 obtained in Chapter 7 give
™^(0,2 — l).10 3 m r
Thus a characteristic feature of the accumulation process was the forma-
tion of larger bodies or planet "embryos" whose masses were far greater
than the masses of the other bodies. In Chapter 11 it will be shown that the
inclinations of the axes of rotation of the planets are due to the randomly
oriented impacts of large bodies falling on the planets. It has been possible
to estimate the masses of the largest of these from the degree of axial
inclination (Table 12). For the Earth it was found that /n/m^lO 3 . Expres-
sion (7) then yields the value 6= 5, which agrees well with the data for 9 in
106
Table 7. Since estimates for mlm 1 based on axial inclination are fairly
reliable, (7) can be used for an independent evaluation of 9. We recall that
in deriving (7) we adopted the simplified relation (1): we assumed that
r+r'^r and m+m'^m . More detailed treatment leads to a cumbersome
expression instead of (7) from which the above value of m/m 1 is obtained for
a somewhat larger 6.
The limiting ratio mjm could only have been actually reached for bodies
m t that had remained over long periods within the zone of the largest body m,
which controlled the relative velocities v of the bodies according to (7.12).
As mlm 1 increased the orbit of m became more and more near- circular and
the supply zone of m comprised an annular region of width 2AR determined
by the mean orbital eccentricity e of the main mass of bodies. If R is the
orbital radius of m, then
u) ai r 3D
All bodies with semimajor axes lying within the interval R±\R could have
collided with m. The distance to the libration point L x of the body m i
is more than one order smaller than AR. In practice, therefore, the zone
of gravitational influence of m did not extend beyond the supply zone. Such
a body m traveling along a nearly circular orbit and having a mass substan-
tially greater than that of other bodies inside the zone can be called a
planetary "embryo." As long as the bodies remained small, velocities and
eccentricities were small and the zones 2AR were narrow. Outside the zone
under consideration lay others in which velocities were determined by the
largest bodies inside them, which, in turn, grew relatively more rapidly
than other bodies, their orbits tending to become circular. Initially, there-
fore, there were many planet "embryos." As long as they occupied different
zones they were not affected by the law that the largest of all should grow
relatively more rapidly, but due to random factors their masses could have
varied by several times. As the bodies grew, so did the velocities, and the
zones broadened. Where adjacent zones overlapped, velocities equalized
and the smaller embryo began to grow more slowly; but it continued along a
nearly circular orbit for a long time, as the distance AR' between embryo
orbits for which the smaller of the embryos is susceptible to strong pertur-
bations from the larger is only three to five times greater than the r L of the
embryo m and is several times smaller than AR . Fusion of the embryos
took place only after m had increased by 1.5 — 2 orders of magnitude. Within
this time the mass ratio must have increased considerably. The gradual
reduction in the embryo population due to mass increase continued until all
the surrounding material had been used up and the distances between embryos
had become so large that mutual gravitational perturbations were unable to
disrupt the stability of their orbits over large time spans. This is the prin-
cipal condition governing the law of planetary distances.
In the last stage of growth the process of accumulation was significantly
more complex, since the increase in relative velocities caused fragmenta-
tions in collisions to play an important role. Collision between embryos and
107
even the largest bodies m 1 would not lead to their disintegration. When
m 1 falls on m, the kinetic energy of the impact per unit mass of m l
4+^=^0 +i) do)
is only one tenth greater than the potential energy at the surface Gmfr. Dur-
ing impact a considerable fraction of the kinetic energy changes into heat.
The remaining energy is expended in ejecting from the area a crater of mass
far in excess of m lm Therefore the kinetic energy of dispersion per unit mass
is on the average far less than the potential energy at the surface of m, and
the ejected material cannot leave the embryo m.
For other bodies collisions are more dangerous. The impact energy of
the body m' when it falls on m x is given by
m ,^ + ^ =ro ,^ L ^ +l)=(29+1)m ,^ (11)
For m' close to m l the impact energy is considerably larger than the total
potential energy of the colliding bodies. In this case collision will lead not
to fusion but rather to destruction of both bodies, to their disintegration into
many fragments. If one half of the impact energy changes into mechanical
energy of dispersion, then for 6 = 5 disintegration will occur when m'> 0.15m!;
if one third of the impact energy so changes, disintegration will occur for
m'> 0.25 m!. Bodies m x could have grown only by collision with bodies of
considerably smaller mass. When allowance is made for fragmentation,
therefore, the limiting ratio mlm x should be greater than the value given by
expression (7). At first a smaller embryo entering the zone of a larger
embryo m will merely lag behind it in growth and will not disintegrate in
collisions with other bodies. But as the growth lag widens collisions become
increasingly perilous and it may be destroyed before it collides with the
largest embryo m.
Using the limiting theorems of probability theory and assuming an inverse
power law of mass distribution with exponent q < 2, Marcus (1967) calculated
the mathematical expectation of the masses of the largest bodies and conclud-
ed that there was no marked gap between the masses of the largest body and
those of the next largest ones. He obtained m/m^3 for q = 3/2. Marcus
seeks to circumvent the difficulties with planetary rotation that arise here by
supposing that the bodies fell on the planet with a velocity far less than the
parabolic velocity at the surface of the planet. The mathematical expecta-
tion of the mass ratios of the largest bodies m k /m k+l is also easy to find
without the cumbersome apparatus of probability theory, that is, directly
from the size distribution function used for the bodies (Safronov and Zvjagina,
1969), The ratios obtained by Marcus will then emerge for the particular
case where the mass of the largest body m=2— q. The power law of distribu-
tion, however, does not take into account the growth characteristics of the
largest bodies (considered above) and gives a poor description of their size
distribution. The assumption that impact velocities were small is strange,
to say the least, as there were no forces capable of significantly slowing the
motion of bodies inside the gravitational field of the planet.
108
27. Accumulation of planets of the Earth group
From the foregoing it emerges that, despite the complexity of the accumu-
lation process and the fact that fragmentation among colliding bodies was
important, the process of growth of the largest bodies (the planetary "em-
bryos") can be described quantitatively in an entirely satisfactory manner if
we assume that their growth resulted from the settling on them of significant-
ly smaller bodies and that they were not fragmented during these collisions.
We can also assume that they moved at all times along circular orbits p=p
situated in the central plane of the cluster where the density of matter is
maximum. The function p (z) inside the cluster can be taken in the form of
the barometric formula (3.12) derived for gases.
The mass increment which the embryo acquires when it uses up other
bodies can be written in the ordinary form
dm
It
= *r;p y 1
(12)
where nr'j is the effective collision cross section and v the mean velocity of
the bodies relative to the embryo, i. e., in practice the velocity relative to
the circular Kepler velocity at the given distance from the Sun. For bodies
with gravitational interaction r t ^l, where / is given by (1) and (2).
As relative velocities v increase, so does the uniform thickness H of the
cluster, and the density p decreases. According to (3.5) the product p t> is
independent of v here and is determined only by the surface density O (Safro-
nov, 1954).
In Chapter 7 we saw that while the mean relative velocities of bodies of
different masses are not the same (different 0), these differences are small
and it is possible to speak of a mean velocity of the entire set of bodies. The
surface density o of matter drops owing to the exhaustion of material by the
planetary embryo m. For a closed planetary zone that does not exchange
material with other zones, one can write
s(t-f). < 13 '
where <x is the complete (initial) surface density, including the embryo m y
and Q is the present mass of the planet.
The expression (12) for the rate of growth of the planetary embryo can
now be written as
dm _ 4i t (1+26) / m\ ( 14)
dt — p °°v q)
This expression differs from Shmidt's (1945) well-known formula for the
rate of planetary growth by a factor 2(1 + 26). The factor (1+2 6) comes
from the increase in effective collision cross section compared with the
geometric cross section due to gravitational focusing. The numerical
factor 2 stems from the fact that in Shmidt's formula, which was derived
by different means, when evaluating the collision frequency only motions
along the z direction were taken into account. It was assumed that within
the time P of revolution around the Sun, any body will intersect the orbital
plane of the planet twice, the probability of its falling on the planet being
109
equal both times to the ratio of the planet's cross section nr 2 to the area of
the planetary zone Q/a .
From (14), on the basis of (7.82), we obtain the obvious relation
dm
dt
(15)
where t\ is the expectation value of the time preceding collision
with the growing planet for a body traveling randomly in its zone. It is
essentially the characteristic time of exhaustion of the planetary material
in its zone.
Let us set
Then (15) becomes
— — 7 ->
Q ~ 2
zV = (*^ h * T (16)
3v£ = l-* < 17 )
where t is the limiting value of x* when m^Q.
If the mean embryo density 6 and parameter 8 are taken to be constant,
then t = const and the above expression is easy to integrate (Shmidt, 1945):
The planet mass m tends asymptotically to Q, while the amount of
material not used up within the zone Q—m decreases exponentially with time.
In the concluding stage of growth where m^^Q and tJ«T , we obtain, from
(15),
Q — m = (Q-m )e-v-<oV\ ( ig j
where Q— m is the amount of unused material at the instant * . For the
Earth zone for = 3 and 6= 5.52, t = 17 million years, while according to
Shmidt' s formula it should have amounted to about one quarter of a billion
years. Within the first billion years of its existence, the Earth had ex-
hausted all the material in its zone. Thus it is entirely out of the question
to estimate the Earth's age from the residue of unused matter in its zone
with formula (18). The interplanetary material that falls on the Earth today
is not a residue of the primary substance of the Earth zone; it is the product
of the disintegration of comets and asteroids continuously entering the
Earth's zone from parts of the solar system more distant from the Sun. The
Earth's age must be evaluated by more direct methods. Thus according to
measurements for rocks, meteorites and chemical elements, it is now
estimated that the planets are approximately 5 billion years old, i. e., nearly
as old as the Sun. A recent estimate, for example, is that of Tilton and
Steiger (1965), who obtain the figure 4750 ±50 million years for the age of
the Earth from lead isotope ratios in ancient rocks of the Canadian shield.
In the derivation of formula (18) for growth it was assumed that the
planetary zone was closed, or more precisely that the total amount of solid
110
material in the zone was conserved at all times and that its initial mass was
equal to the present mass of the planet (relation (13)). This is an important
assumption, and for the giant planets it does not hold, since in the final
stage of growth their source zones fused into one open zone (bodies were
ejected beyond the solar system). But for planets of the terrestrial group
(with the exception of Mars) it is wholly applicable. Bodies with relative
velocities corresponding to = 3 could not have traveled beyond Mars' orbit,
for example, and most of the bodies never reached it.
Formula (18) makes it possible to calculate the duration of the process of
planet formation. Owing to the asymptotic character of its vanishing, the
choice of concluding instant for this process is arbitrary. In 1954 and 1957,
we estimated the growth span up to the instant when a planet reaches 97% of
its present mass, i. e., when z = 0.99. For z = 0.99 the right hand side of
(18) is given by / (0.99)= 6.0 and thus the planet formation time x p as deter-
mined in this manner is given by
x , = 6 V
(20)
For 6 = 3 and 6= 5.52 the Earth's growth time is 10 8 years. The variation
in the mean density of the planet during the growth process can be allowed
for fairly accurately (within 1—2%) by taking the average of the initial and
final mean planet density for 6 when calculating t in (16). Then for the
Earth one can take §^4.5 and r p = 0.88 • 10 8 years. Within 100 million years
the Earth's mass must have grown to 98% of its present mass. A graph
showing the rate of growth of the Earth' with allowance for the variable 6 is
given in Chapter 14.
Table 11 gives the characteristic exhaustiontimes t of planets of the Earth
group (in million of years), as obtained for present values of planet mass
and density and for 6 = 3 and 6 = 5, in accordance with (16). The surface
density a was determined from present planet masses; boundaries between
planet zones were taken to be the average of figures obtained by Shmidt and
by Gurevich and Lebedinskii from laws governing planetary distances.
TABLE 11
Mercury
Venus
Earth
Mars
°o
M8 = 3)
M9 = 5)
1.5
10
6
16
5
3
10
17
11
0.3
400
250
The value of ct obtained for Mars is unusually low (0.3g/cm 2 ). Assuming
that the width of the zone was 2AR , in accordance with (8), then for 6=3,
°o^ 1 g/cm . Since within the Jupiter zone a surface density of solid matter
of 20— 30 g/cm amounts to 3— 5 g/cm 2 when translated to silicates, while in
the Earth zone ct « 10 g/cm 2 , one would expect that within the Mars zone it
would be - 7— 8 g/cm 2 . This means that only about 10% of the solid substance
in its zone was absorbed by Mars, and the remainder left the zone. Conse-
quently, it becomes meaningless, strictly speaking, to evaluate t for Mars,
111
as the material in its zone was, for the most part, not used up by Mars.
What happened to Mars is nearly the same as what happened to the asteroids
(see Chapter 12). Owing to external perturbations (influx of bodies from the
Jupiter zone; see Section 34), velocities in the Mars zone rose far more
rapidly than could be expected if the perturbations had been due to Mars
alone, and this slowed down its growth very markedly (in expression (2) for
the collision cross section, (l + 29)-*l).
Since t p cc x oc P } planet growth was on the average more rapid in the inner
portion of the cluster than in the outer cluster.
For z <^ 1 the growth formula (18) is much simpler:
, = 3« o = 3(-0\. (2D
At the early stage the density drop due to exhaustion is insignificant and
the radius r increases in proportion with time. Introducing m — a 3 /p* 2 in (21)
in accordance with (6.6), we obtain the time t (m^ necessary for the mass of
a small body to increase to that of condensations formed by gravitational
instability. It is independent of a and is proportional to R'^. Even in the
Mars zone it amounts to only 3-10 years (see below):
Zone Mercury Venus Earth Mars
M m o), years 3 • 10* 2 • 10 3 8 • 10 3 3 • 10 4
Consequently, even if any factor prevented the appearance of gravitational
instability in the region of the Earth group planets (see Chapter 3), within a
fairly short time direct particle growth will have led to the formation inside
this region of bodies with dimensions of the order of kilometers. Low rela-
tive velocities practically ruled out the possibility of particle fragmentation
in collisions. An extremely low gas density (10~ 9 g/cm and less) made
possible efficient adhesion, as in cold welding (Levin, 1966).
112
Chapter 10
ROTATION OF THE PLANETS
28. Critical analysis of earlier research
The problem of the planets' rotation is one of the most difficult in plane-
tary cosmogony. Not unexpectedly, different cosmogonic theories envisaged
different solutions. An extensive review of work on planetary rotation was
published in 1963 by Artem'ev. Below we consider only the more important
theories.
Authors developing the Laplacian hypothesis usually related the planets'
direct rotation to the action of the Sun's tidal forces. Poincare gives the
following schematic description of this process (1911). A gaseous ring
separates from the central condensation and begins to revolve around the
Sun as a rigid body owing to frictional forces. Eventually it becomes
unstable and disintegrates. Separate sections of the ring begin to move
along circular orbits. When two clusters situated at somewhat different
distances from the Sun combine, retrograde rotation sets in, since the inner
cluster will travel more rapidly along the orbit than the outer one. However,
the tidal forces of the Sun attract the cluster and impart to it a direct rotation
of period equal to the period of revolution. Contraction of the cluster due to
cooling reduces the tidal forces and increases the rate of rotation.
None of these conjectures is admissible today. Friction inside a ring
revolving around the Sun would not cause it to revolve as a rigid body. It
would merely move the inner part of the ring closer to the Sun and cause the
outer part to move away, rotation remaining Keplerian at all times. The
idea of the disintegration of the protoplanetary cloud into small gaseous
clusters and their subsequent fusion is equally untenable. Such clusters
would be unstable, tending to disintegrate rather than combine. The only
way around this difficulty is by having dust — not gaseous — condensations.
But dust condensations are small and contract so rapidly that tidal forces
could not markedly affect their rotation.
In the planetesimal hypothesis of Chamberlin (1904) and Moulton (1905)
it was assumed that the planet acquired direct rotation in the process of
growth as a result of the influx of planetesimals. The authors supposed that
for the main the planet acquired positive rotational angular momentum only
from bodies with the perihelial distance R Q < R p < R 9 +r and bodies with the
distance R — r < R a < R at aphelion, where r is the radius of the growing
planet and R its distance from the Sun. It was found that, for an orbital^
eccentricity e = 0.2 for the bodies and impact parameter equal to 6.5 ■ 10 cm,
bodies of this class could have imparted the required rotation to the Earth
provided their mass was equal to 5.7% of the Earth mass. However, it was
113
not made clear whether such a high percentage of bodies within such a
narrow range of perihelial and aphelial distances is possible.
Hoyle (1946) suggested an explanation for planetary rotation based on
the concept of planet growth by accretion (aggregation; see Section 1) of
diffuse matter envisioned as a solid medium. The probable capture radius
for a planet of mass m and radius r was taken to be half the distance at
which the tidal force of the Sun equals the planet's gravitational attraction:
r a — -j(ml2M Q f 3 R, The rotational momentum IK imparted upon accretion by
substance Am was taken to be 2/5 a>r 2 a Am (spherically symmetric accretion),
where a>=<o fl /4 (see Chapter 6). The period of rotation was found to be
3— 4 hours, and would be even less if allowance is made for the concentration
of matter toward the center of the planet. Hoyle accepted this very high
speed of rotation under the influence of Lyttleton's view of the rotational
instability of primary planets and the separating away of satellites. Lyttle-
ton (1960) still maintains this view, but it lacks a firm foundation. On the
other hand accretion could have played an important role only in the growth
of Jupiter and Saturn, when these had become massive enough to absorb
gaseous hydrogen. Recently Hoyle (1960) renounced the application of the
accretion mechanism to Earth group planets. If the capture radius in
accretion is revised down to 2—3 times less than the value adopted by Hoyle,
the interpretation of Jupiter's and Saturn's rotation in terms of accretion
theory will become satisfactory.
The theory of planet formation from massive protoplanets (Kuiper, 1951;
Fesenkov, 1951) postulated the formation of massive clusters as a result of
the onset of gravitational instability in the gaseous component of the cloud.
The condensation of the protoplanet was envisioned as a process of collection
of material along the orbit of the primordial cluster. To determine the
planet's rotation Fesenkov computed the angular momentum of a section of
a torus of diameter 21 with reference to the axis passing through the center of
the section. It was found that the present planetary rotation is obtained only
for values of / far smaller than the width of the planetary zone. (The torus
diameter should be 70 times smaller than the zone width in the Jupiter zone
and 300 times smaller in the Earth zone.) The excessively high planetary
rotation yielded by the theory of massive gaseous protoplanets further com-
pounds the serious difficulties which it presents (Ruskol, 1960) and which
have defied solution.
Gurevich and Lebedinskii (1950), in their approach to the problem of plane-
tary rotation, considered the rotational angular momentum of dust condensation
whose fusion led to the formation of the planets. They obtained an expression
for the angular momentum of the condensations in the form fc M m {MJM ( 7 ) )* J
where k is the specific orbital angular momentum and M n the planet mass.
From this it was concluded that the rotational angular momentum of a planet
should be equal to the orbital angular momentum multiplied by a certain
function of the planet mass. This result was illustrated by an empirical
dependence which fits all planets other than Saturn and Neptune. This
conclusion essentially rests on the implicit assumption that condensations
combining with each other suffer central collisions. In noncentral collisions,
in addition to the proper rotational angular momenta of the combining conden-
sations, one must also take into account the considerably greater angular
momenta associated with their relative orbital motion.
114
Looking at the problem of planetary rotation, Shmidt (1957) started with
an analysis of the general laws governing the process of fusion of material
into a planet. He wrote down conditions of energy and angular momentum
conservation for transition from a particle cloud to a planet. The angular
momentum of the particles situated in the planetary zone changes into the
orbital and rotational angular momenta of the planet. The smaller the
orbital angular momentum, i. e., the smaller the radius of the planet's
orbit, the greater its rotational angular momentum should be. But as the
orbital radius of the planet decreases, so does its orbital energy while as
a result, according to Shmidt, the thermal loss of energy in the process of
planet formation increases. Hence Shmidt' s major result: since the energy
losses in this process are large, the planetary rotation should be direct.
The mathematical formulation of this result reduces to the following.
Consider a cloud of particles traveling around the Sun along circular
orbits lying in the same plane. From these particles is formed a planet of
mass m and orbital radius R . Let R, and R m be the mean distances between
cloud particles averaged over the energy and the angular momentum
respectively:
] R dR *,
_!___.*. J_ f y (R) dR
*. ** "~m J R '
j <p (R) dR *,
j ^Rf (R) dR ft,
\v{R)dR
».
where <p (R) is the mass distribution function of the particles over the
distance from the Sun, and i?j and R 2 are the boundaries of the zone of the
planet under consideration.
One can prove that R m > R 4 at all times (just as the mean square is always
greater than the simple mean). Therefore if the condition
*o<*. (2)
holds, the following inequality should be satisfied:
«»<«„. (3)
i. e., the rotation should be direct, since the planetary orbital momentum,
proportional to \JW , is smaller than the angular momentum of the cloud
particles, which is proportional to s[R~ m .
The conditions under which relation (2) is fulfilled were not analyzed.
Shmidt believed that "we cannot determine the sum of these losses quanti-
tatively, but there is no doubt that the losses are large." He proposed
further that these same factors were responsible for the direct revolution
of most of the planetary satellites and that the retrograde revolution of
distant satellites was due to nonfulfillment of condition (2).
115
Let us analyze more carefully Shmidt's equations for the energy and
angular momentum balance and his condition for direct planetary rotation
(Safronov, 1962c). We will adopt the following notation: U Q — potential
energy of planet with reference to the Sun; U p — potential energy of planet
as a sphere; U c — potential energy of particle interaction; E — orbital
energy of planet (potential plus kinetic); E r — kinetic energy of planetary
rotation; E t — loss in energy of mechanical motion resulting from its
transformation into other forms of energy, such as warming, radiation,
phase transitions, etc.; M — Sun's mass; m — mass of growing planet; R —
radius of planetary orbit with reference to Sun; K — orbital angular momen-
tum of planet; K r — rotational angular momentum of planet. The balance
equations derived by Shmidt then become
<jGM\s/R 9 (R)dR=:K + K ry (5)
where
^° = — TS7' K =zm\/GMW^ m=,\ f (R)dR. (6)
*i
According to Shmidt, the first term in the left-hand side of (4) represents
the sum of the kinetic and potential energy of particles moving round the Sun
along circular orbits, while the term in the left-hand side of (5) represents
their total angular momentum with reference to the Sun. These terms were
written down for the Sun's gravitational field alone and do not allow for
gravitational fields induced by particles.
Artem'ev (1963) has pointed out that these balance equations fail to take
account of the energy and angular momentum of the Sun, which after the
planets were formed ceased to lie at the center of gravity of the system. Yet
the orbital angular momentum of the Sun is several times greater than the
rotational momentum of a planet. In (4) — (6) E and K were taken for the
planet's motion with reference to the Sun. In reality one should take the sum
of E and K for planets and Sun with reference to the center of gravity of the
system. In the presence of numerous planets the Sun's motion becomes very
complicated. Rigorous balance equations would have to be written down
simultaneously for all bodies in the solar system, and they could not yield
concrete results for isolated individual planets. But analysis is possible if
one confines oneself to a single planet traveling around the Sun. Denoting by
E c and K et respectively, the sum of the orbital energy and the sum of the
orbital momenta of the planet and Sun, with reference to their center of
gravity, it is easy to find that
p GMm
E > = —25T<
K, = ml/Z^K. < (7)
* r 1 + m(M *
where R is the distance of the planets from the Sun (not from the center of
gravity). Quantities E g and E Q are identical while ^differs from K > given
in (6), by an amount «^ — y-^-^o-
116
Corrections to the left-hand side of equations (4)— (5) due to allowance
for the gravitational field of particles scattered throughout the planetary-
zone and having total mass m will be of the same order. In the first equa-
tion the correction to the kinetic energy will hem — UJ2. Combining it with
U c , we denote the total correction by K~m~Eo- The correction to the left-
hand side of the second equation will be denoted by ~2jjfK<>. Obviously,
Xj~X 2 ~ 1. Then, replacing the integrals in accordance with (1) by R t and
R m respectively, we obtain
where
iH 1 — - x .-f)£. ( 8 )
^: = [l+A-i(l+X 2 )^-]Vft 0) (9)
2/i
• = % + «« + «, = SBfc(tf, + tf, + £ P ) ( (10)
k = -
Eliminating i? , we obtain
2 *~*=^+'+^. (11)
where X 8 =l+X 1 -r-X,~ 1.
The ratio ((R m —R.)tR, depends chiefly on the width of the source zone and
only weakly on the form of q? {R). For the Earth zone it is of the order of
10~ 2 . There is no basis for assuming that e < (see below). In precisely
the same way, X s > 0. The right-hand side is found to be positive and thus
the rotation must be direct. Formally the problem would appear to be
solved. In reality this is not so, as expression (11) yields an inadmissibly
high value of k. For the Earth it gives a value which is 10 4 times faster than
the actual rotation (the ratio of rotational to orbital angular momentum for
the Earth is given by A»3 • 10" 7 ).
The reasons for this result are to be found in a defect of the scheme
itself. It seems natural to suppose that the particles traveled originally
along circular orbits. For small body masses gravitational perturbations
must have been weak and the particles moved along orbits that were nearly
circular. As the planet grew the deviation of body orbits from circular
orbits increased, and all bodies inside the zone were able to combine into
a single planet. The balance described above would be valid if the planetary
zone remained closed at all times. That this supposition is inadmissible is
shown by the exceedingly fast rotation one then obtains. As eccentricities
increase owing to encounters between bodies and planet and among the bodies
themselves, some bodies travel out beyond the outer boundary of the zone
and remain there (becoming "stuck"), removing excess momentum from the
zone; others cross the inner boundary, removing momentum which is less
than the mean momentum. A simultaneous influx of bodies takes place into
117
the planet zone from the outside. These exchange processes are not
compensatory and as a result the total angular momentum of matter inside
the zone, as well as the total energy, including thermal losses, does not
remain constant.
The statement of the problem becomes more valid if we allow for the
eccentricites and inclinations of the orbits of bodies and particles and
include in the function q> (a) only those that actually fall on the planet. The
difficulties stemming from the fact that the zone is open can then be circum-
vented to a large extent by considering not the total balance between initial
and final states but rather a "differential" balance, i.e., the balance for
specified values of m, e and i and for growth of the planet mass by a small
quantity Am. The exchange of angular momentum and energy between bodies
landing on the planet and bodies that do not land on it may especially be
disregarded, since it is this exchange that causes the bodies to acquire the
orbital eccentricities e and inclinations i, which we are taking as initial
data. The influence on the planet's motion of bodies not landing on it is
apparently negligible, although it is not inconceivable that while traveling
along its circular orbit the planet experienced very limited retardation, and
therefore reduction in R, due to encounters with bodies whose centrode
velocity, as is known, is slightly smaller than the circular velocity.
Let us assume for simplicity that the orbital eccentricity and inclination
to the plane of the planet's orbit are the same for all bodies and particles.
Quantities a, e and i will be understood to be mean unperturbed elements
describing the motion of a body that is not in the process of encountering
other bodies. Let the mass imparted by bodies and particles with orbital
semimajor axes lying between a and a + da to the planet be Am? (a) da. Then
[<?{a)da - 1. Further, let the planet move along a circular orbit ( a = /? ).
Instead of equations (4) and (5) we obtain
Am j sja (1 — e 2 ) cos i f(a)da — A(m\fa^)^ 9 (a) da X
*> a,
We further have
where
^H 1 -^' A(m V T ) = (l+-I)^m,
din at,
1 dlnm •
Next, applying notation (1) to the orbital semimajor axes instead of the
orbital radii, we obtain the following expression from (12), by analogy with
(9):
118
where
^„(l- e 2 )cosi = ^{(l + 2)[l_| ( i + x 2 )g_fcj, (13)
•'«&£<*, + *. + *.). "=7=^- < 14 >
VGMa
Eliminating <z , we find that
a„(l _ e *)cos 2 i = [(l - T )(l _\,£)_ e '] X
xtO+Dl'-ld+^M 2 *- (15)
The quantity 7 characterizing the variation in the orbital radius of the
planet embryo can be evaluated from the second equation in (12), setting
9 (a)=ca~ n and A# r = owing to its smallness. Calculations indicate that for
admissible values of n the value of y is of the order of e 2 . In (15) y yields
only terms of the order of e 4 and higher powers. Retaining only infinitesi-
mals of the second order in e and i and first order in k' and mIM, we obtain,
by analogy with (11),
^^2 + ;2 + 2 *'-s'-X 3 -£. (16)
Comparing (16) and (11) we see that when the eccentricites and inclinations
of the orbits are taken into account, there appear additional terms e 2 and i 2
of the same order as the rest. Hence they make a substantial difference to
the result. Consequently a scheme based on circular particle orbits is
clearly inadmissible.
To determine the left-hand side of (16) we introduce the dimensionless
distance
and express the distribution function <p (a) in terms of x, retaining only
second order terms:
<? = fl (l + Cl x + c^). (17)
Introducing this expression for <p into (1), where R should be replaced by
a throughout, we can find a m and a e . Calculations show that up to the third
order in x expression (16) will be independent of c x and c 2 :
^=4(l-* 2+ ...). (is)
The value x % corresponds to the outer boundary of the zone (a 2 ). For the
present-day Earth one can take £ 2 ?^0.6 and e^0.2. Since
«>=£-.. **=!#;. (19)
119
for small e
o., — a, 2e
Introducing this value into (16), we obtain
J^S + P + W-^ (20)
or
e; + e;~-u; + ±.v*[$* + i*] + w*-%-%.vi < 21 >
This expression can be written in more explicit form by introducing the
body velocity V (with reference to the planet) unaffected by the latter' s
gravitation (velocity before encounter). The velocity component perpen-
dicular to the plane of the planet orbit v^iV c> where V 9 is the circular
velocity and the component in the orbital plane v Rlf z&eV 9 - y \ — -r-cos 2 cp , where
cp is the angular distance of the body from perihelion at the instant of encoun-
i;*«(X 4 c* + O^I (22)
ter. Let ^ = V 2 ^* Then
and from (21) we find that
(23)
The exact value of X A is difficult to compute. It requires knowledge of the
density distribution 9(a) inside the cluster. In any event X 4 is close to 3/4
and the next to last term in (23) is at least one order smaller than v 2 . If one
takes the simple mean of (1 — -^-cos 2 cpV then ^ 4 = 5 /8 an d ^ ne next to last term is
equal to y 2 /16.
Analysis of this equation leads to the following conclusions.
1. The fusion into a planet of bodies traveling along elliptical orbits does
not lead to the inadmissibly rapid rotation which is obtained for the fusion of
bodies moving along circular orbits.
2. To order of e 2 (the accuracy with which expression (23) was obtained)
the thermal losses incurred when particles strike the planet surface are
equal to the sum of the potential energy released in the fall and the kinetic
energy of the particle before encounter with the planet ( E' r is three orders
smaller than U' p ), This is the value we took for the losses when evaluating
the Earth's primordial temperature (1959).
3. Expression (23), which was derived from the balance equations, con-
tains two unknown quantities: the velocity of rotation of the planet and the
thermal losses in the accumulation process. The balance equations are not
sufficient for solving the problem of the planets' rotation. Only by analyzing
a definite collision mechanism will it be possible to determine both rotation
and losses with the aid of these equations.
120
4. If the term E' r in (23) characterizing the energy of rotation had been
significantly smaller than the term k'V] , then # would have increased with
increasing E' t and one could have expected direct rotation with high thermal
losses. The term with k' is the smallest in (23), and it is practically
impossible to evaluate it, as it appears in (23) among quantities which are
four orders larger. Consequently, no conclusion regarding the direction of
planetary rotation can be drawn from (23). For the present-day rotational
velocity of the Earth the term k'V 2 e is 2.5 orders smaller than E' r . The
relationship between rotation and thermal losses should therefore be inferred
from E' r and E' t rather than from k' and E\. These quantities appear through-
out as a sum. For specific eccentricities of body orbits, the faster the
planet's rotation, the smaller should be the thermal losses in the accumula-
tion process. This result is qualitatively understandable: acceleration of
the rotation intensifies as the number of particles striking in the direction of
rotation increases, and as the number striking counter to the direction of
rotation decreases (i. e., as the mean velocity of particle impacts and hence
the thermal losses decrease). Incidentally, this result also holds in the
case considered by Shmidt of body motion along circular orbits. In his
balance equations the energy loss also appears as a sum together with the
rotational energy.
Thus when one allows for the planet's rotational energy, which in the
balance equations plays a considerably larger role than the planet's rota-
tional momentum, one is led to conclude that thermal losses decline as the
velocity of rotation increases. The rotational energy seems to arise at the
expense of thermal losses.
It is possible to determine what planetary rotation corresponds to maxi-
mum thermal losses. Let us assume that a certain set of particles and
bodies combines to form a planet in two ways characterized by different
thermal losses. In the first case the planet is formed on a circular orbit of
radius R and in the second, on one of radius R+6R. The total angular
momentum should be the same in both cases, and therefore changes in the
planet's orbital momentum are compensated by changes in its rotational
momentum
ZK, = -8* = — =- Y^- * R >
which leads to a change in the rotational energy
E r = 1 iy r ; K r = 7 r o> r ; ZE r = 7 r u>> r = a> r 8tf r ;
The change in orbital energy
will be much smaller than 6E r in view of the smallness of the orbital angular
velocity a> e compared with the rotational velocity a> r . Therefore the change
in overall mechanical energy
121
^o + ^ = -f^K-%)S/? (24)
is practically determined by the change 6E r . Next,
E -\-E r + E t = const; lE t = ~^hE — B£ r .
Therefore if the thermal losses increase (i. e., if the mechanical energy
decreases), for (o f > o) c one should have 6R > and the velocity of rotation
will decrease.
Energy losses are maximum when the sum of the orbital and rotational
energies is minimum. For this to happen it is necessary that &E -\-6E r = 0,
or, according to (24), co^co,.. Consequently, thermal losses are maximum
when the planet rotates about its axis with the velocity of rotation of the
cluster itself, i. e., when it does not rotate relative to the cluster. For
maximum thermal losses the rotation is found to be direct but excessively
slow compared with actual rotation of the planets. The rotation relative to
the cluster increases with decreasing thermal losses. From the standpoint
of the loss it is unimportant in which sense the rotation proceeds if the
latter is reckoned from o) tf , since it can be shown that the losses are the
same for the rotational velocities w { +Aw and (o c — Aw .
An attempt to attribute the planets' direct rotation to large thermal losses
during their formation was also made by Khil'mi (1951) based on a few
general relations of mechanics. However, a closer analysis of these
relations by the same author (1958) indicated that it does not inevitably
follow from them that the planets' rotation was direct.
It should be stressed that the objections mentioned above do not affect the
foundation of Shmidt's theory regarding the formation of the planets by the
accumulation of solid bodies and the considerable role of thermal losses in
this process. If there had been no thermal losses collisions between bodies
would have been absolutely elastic and their accumulation into planets would
have been impossible. Our objections are directed only at the idea that the
planets' direct rotation is also due to large thermal losses. In reality both
the planets' direct rotation and a certain loss of energy in the process of
accumulation were due to concrete conditions of collision between combining
bodies, i. e., to the basic laws governing their motion. If the planets had
formed in a nonrotating cluster, for example, they would not have acquired
regular rotation although losses would have reached values of the same order.
A substantially different explanation of the planets' rotation was suggested
recently by Artem'ev and Radzievskii (1963, 1965). The authors conjecture
that the planet acquired nearly all of its rotational momentum not from bodies
falling on it directly, but from bodies originally captured by the planet as a
result of inelastic collisions in its gravity field, and subsequently falling on
it. In one variant the velocities of the bodies after collision were assumed
to be equal to the circular Kepler velocity; in another variant they were
taken to be equal to the corresponding rigid rotation of a Hill region around
the Sun. This idea of a two- stage process of fall on the Earth enabled them
to interpret r in the expression for the angular momentum acquired by the
planet ( y m^dm in the first variant; see Chapter 6) not as the planet radius,
but nearer the radius of the largest closed Hill surface (i. e., at least two
5079 122
orders greater). It was found that for the planets to acquire their present
angular momentum it is necessary that such capture be experienced,
depending on the scheme, by a few percent of the mass to nearly the
entire mass of the planet (the latter would seem to be more probable). This
is much more than the mass of all the satellites, which according to modern
views were formed from material in the same planetary cluster captured by
the gravitational field of the planet as a result of inelastic collisions among
bodies and particles in its vicinity (Ruskol, 1960). Artem'ev and Radziev-
skii's hypothesis thus leads to important consequences regarding the charac-
ter of the formation process and the evolution of satellite clusters around
the planets. To test the hypothesis one would have to calculate directly the
amount of material captured by the planet and its angular momentum with
reference to the planet. This in turn requires further progress in accumu-
lation theory.
29. Methods for solving the problem
Of the two conservation equations (4) and (5) employed by Shmidt, only
the equation for angular momentum conservation is directly related to the
problem of the planets' rotation. The equation of energy conservation only
makes it possible to evaluate thermal losses in the process of accumulation
after the planet's rotation has been determined from the angular momentum
equation. Since the planet's rotational momentum amounts to only one
millionth of its orbital momentum, the form (5) and (12) of the angular
momentum conservation equation is extremely inconvenient to use in
quantitative estimates. It is far more expedient to write the equation in
a coordinate system directly related to the planet.
Consider the collision with a planet of mass m and radius r of a particle
Am having, at the instant of impact, a velocity v' with reference to the
planet, directed at an angle to the inner normal at the point of incidence.
Its total angular momentum at impact is conserved with reference to the
center of gravity of the system (m, Am). From this conservation law we
obtain the increment in angular momentum of the planet when struck by
the body Am;
A jr mAm , . a mAm , . . . A mAmv'r . fi trie \
* K r = m ■ Am vr e Sln & ~\ nr~ v* ( r — r.) sin 6 — — r~r~ s "i 9, (25)
m ~\- Am c ■ m + Am x c/ m + Am ^ '
where r c is the distance of the center of gravity after incidence of Am from
the original center of the planet. It can be shown that this equation is equiv-
alent to the second equation of (12) and that it can be derived from it by
simple vector transformations. Since ordinarily Am <^ m, it can be written
in the simpler form
Atf r ^AWrsinO, (25')
i. e., the angular momentum acquired by the planet when the body Aw falls
on it is equal to the angular momentum of Am with reference to the planet at
the time of impact. Thus the problem of planetary rotation can be reduced
to a statistical discussion of a limited three-body problem (Sun, planet,
123
particle of small mass). Each particle imparts to the planet an angular
momentum directed basically at random. The problem is to obtain the
mean of the incidence of many particles. An approximate discussion of the
problem with the introduction of a sphere of action and replacement of the
three-body problem by two two-body problems has been carried out by
Artem'ev.
It seems that any real hope of solving this complex problem lies chiefly
with numerical methods. In-dividual attempts at a numerical solution are
already under way. The results obtained are encouraging, though as yet no
general conclusions may be drawn. Kiladze (1965) has computed quasi-
circular orbits on a computer for a limited circular three- body problem
(the orbits start in the vicinity of the libration point L 3) behind the Sun, and
end at the planet's surface). A one- parameter family of these orbits was
selected with the aid of a condition imposed on the initial coordinates and
velocities, enabling the author to simplify his equation to some extent. The
rotational momentum imparted to the planet by particles moving along these
trajectories was found to be negative for a planet mass m equal to the mass
of Jupiter. Kiladze believes that for m / M Q < 1/1500 the angular momentum
ought to be positive. He has initiated calculations for other classes of initial
particle orbits with a view to estimating the mean rotational momentum
imparted to the planet by the entire set of particles falling on it.
Giuli (1968) has computed several tens of families of trajectories on a
computer, beginning at great distances from the planet in the form of ellipses
with definite values of a and e in each family and ending at the planet's
surface. For most families the angular momentum imparted to the planet
was found to be negative, but the overall angular momentum was positive
thanks to the particularly effective contribution of certain families. Accord-
ing to Giuli, the period of rotation of the Earth calculated in this manner is
7.3 hours. This work is very interesting, but for 15% of the trajectories,
which were more complex, calculations were not carried through and thus
it is not clear how accurate these results are.
For a complete solution of the problem of the origin of the planets' rotation
it is also necessary to evaluate the amount of material reaching the planet
from the satellite cluster and the rotational momentum it imparts to the
planet (see Section 28), One factor leading to settling of this material is the
growth of the planet's radius and mass and the corresponding shrinking of the
orbits of all particles in the cluster; another is the small mean angular
momentum of the captured material. In order for the material settling down
from the cluster to impart the entire necessary angular momentum to the
planet, its mass must amount to a significant fraction of the planet mass.
It seems that the probability for this is low." According to Ruskol, the
* The fact that the orbits of close regular satellites coincide with the equatorial plane of the planet is not
decisive proof in favor of this particular method of acquisition of angular momentum by the planet. The
planet's equatorial contraction leads to precession of the orbits of the particles and bodies in the satellite
cluster with reference to its equatorial plane (Goldreich, 1965a). Owing to differences in the periods of
precession of individual bodies, the cluster, originally characterized on the average by a certain inclination
€ to the planet's equatorial plane, expands so that fairly soon the body orbits lie on both sides of the equatorial
plane (within angle ± e) and the latter becomes the mean plane of the cluster. As a result of inelastic colli-
sions among the bodies the cluster flattens out, but it lies within the equatorial plane. Any deviation from
this plane, such as the prevailing incidence of material in the plane of the ecliptic, will again lead to
thickening of the cluster (with the aid of precession) and its mean plane will again tend to the equatorial plane
of the planet. With increasing distance from the planet, the plane with reference to which the orbital
precession takes place shifts away from the equatorial plane and approaches the plane of the planet's orbit.
124
density of matter was highest in the inner part of the cluster. It is therefore
not excluded that a considerable fraction of the rotational momentum of the
planet was imparted to it by the material of the satellite cluster. It may be
that the problem of the planets' rotation will prove to be closely related to
that of the formation and evolution of satellite clusters.
In the absence of a rigorous solution to the problem of planetary rotation,
interest attaches to qualitative considerations regarding possible laws
governing rotation. Certain results can be obtained by exploiting the concept
of asymmetry of the impacts of falling bodies and particles (Safronov, 1960a).
Within the limits of the two- body problem the rotational momentum imparted
to a planet of mass m and radius r by an individual particle m i can be
written as
#< = P</n,i>r,
where v is the relative particle velocity before encounter with the planet,
p<r=Z, is the impact parameter, and p, lies between and \J1 -f 26 . Despite
the fact that the direction of the vector K, is basically random, due to the
presence of a third body (the Sun) and to the rotation about it of a cluster of
particles the mean value of K 4 is nonzero, i. e., there exists a systematic
angular momentum component. Let us denote it by K t . Then
dK 1 = fivrdm,
where the coefficient P characterizes the asymmetry of the impacts. If we
assume that p remains constant throughout the process of planet growth,
then for y=V^.cxr we obtain
K l cc\ rHm oc m if * oc r 2 m (26)
2
and since K 1 ~j\^r t m , w » const. Then the angular velocity of rotation of the
planet will on the average remain constant during its growth process.
If we assume further that p is independent of the planet's distance from
the Sun we find that the angular velocities of rotation should be equal for all
planets (no allowance being made for the variation of the planet density 6 and
the parameter 9). However simplified this assumption may be, the result
is not very far from reality. It is well known that with the vast differences
in planet masses, the angular velocities of planetary rotation vary compar-
atively little. If we allow for the parameter 8, the inhomogeneity coefficient
of the planet ju and its density 6, which appear in the expression for K, we
then obtain (regarding them as constant throughout the process of planet
growth)
»cc V(2 + t/6)&/|w (27)
For 9 ^> 3 the dependence of u on 9 is very weak and can be disregarded. The
variations in u) due to variation in 6 and /u are small. Quantity u increases
slowly with m owing to the increase in 6 and decrease in^.
The fact that the values of 6 and n averaged over the entire period of
planetary growth (and not the present values) must be taken in (27) makes it
125
somewhat complicated to compare (27) with actual data for the planets.
Moreover, most of the planets have their own characteristic features which
are not taken into account in the above simplified growth scheme. The rota-
tion of Mercury, Venus and the Moon is completely braked by tides. Tides
slow the Earth's rotation substantially and partially brake Neptune's rotation.
Although Triton is more massive than the Moon and its distance from Neptune
is somewhat less than the Moon's distance from the Earth, it, unlike the
Moon, is drawing closer to Neptune: earlier it was farther from Neptune
than it is today. This is why the retardation of Neptune's rotation could not have
been considerable. Jupiter and Saturn contain more gas than solid substance,
but the specific angular momentum acquired in the accretion of the gas may
have been different from that acquired in the accumulation of the solid
substance. Uranus has an obvious anomaly: its axis of rotation is inclined
at an angle of 98°, because its random component of rotation was greater
than the systematic component (see Chapter 11). That is, the systematic
component is simply unknown. Pluto wholly fails to conform to the general
pattern. Its mass, moreover, is still not known. This leaves only two
planets with which to establish the laws governing rotation — Mars and
Neptune. This is clearly insufficient in view of the fact that a planet's
rotation may depend not only on its mass but also on its distance from the
Sun.
MacDonald (1964) constructed an empirical function for the dependence
of specific rotational momentum on the mass of the planets (k(m)), approxi-
mating it by a power function with exponent close to 0.8. Assuming that the
Earth satisfied this dependence primordially, he obtained an initial period
of rotation of 13hrs (this corresponds to an initial distance between Moon
and Earth of about 40 Earth radii). Hartmann and Larson (1967), introducing
asteroids with known periods of rotation, approximated an inclusive depen-
dence k (m) for all bodies by a power function with exponent 2/3, which
corresponds to invariance of the period of rotation (disregarding differences
in 6 and ji). For the Earth this dependence yields an angular momentum
equal to the total angular momentum of the Earth- Moon system. From this
the authors infer that the Earth and Moon originally constituted a single body
with a period of rotation of 4— 5 hrs. A similar function k (m) was obtained
by Fish (1967).
The construction of a single function k (m) for planets and asteroids would
be meaningless if the asteroids' rotation had altered substantially since they
were first formed. According to Hartmann and Larson, collisions between
relatively large asteroids (over 10 g) were exceedingly rare and could not
have altered their rotation appreciably. They see confirmation of this in
Anders' data (1965) on the absolute size distribution of the asteroids, which
inAnders' view show that the largest bodies underwent little fragmentation,
and in remarks by Alfven (1964) to the effect that if the number of collisions
had been large, the rotational energy of asteroids of different mass would on
the average have been the same (equipartition), which is not confirmed by
observation. But Hartmann and Larson's reasoning is not convincing enough.
Anders assumes that asteroids brighter than the ninth absolute magnitude
are "primary;" weaker ones underwent fragmentation. But on the average
velocities of rotation were the same in both groups. Consequently, the pri-
mordial asteroids somehow managed to acquire large rotational velocities
characteristic of colliding asteroids. The tendency to even distribution of
the energy of rotation could only have occurred in a system of bodies with
126
absolutely elastic collisions without fragmentation. In the presence of
dissipative processes the energy of small bodies decreases much faster.
Moreover, in collisions between asteroids with velocity ~ 5 km/sec, the
smaller one should disintegrate and drop out from among those of known
rotation.
Lastly, the special conditions that prevented the asteroids from combin-
ing into a single planet during the process of growth could not have failed to
leave their mark on the asteroids' rotation. In the first place, the large
relative velocities of the bodies in the concluding phase, which caused the
asteroids to stop growing (see Section 34), must also have led to a higher
velocity of regular rotation (9< 1 in formula (2 7) for the angular velocity).
Secondly, owing to bodies flying into the asteroid zone from the Jupiter zone
and to the considerable relative velocities of the bodies inside the asteroid
zone, there was no great difference between the mass of the planet embryos
m and the masses of other large bodies m 1 (see Section 26). This must have
contributed considerably to the asteroids' random component of rotation
(Section 30), and the latter may have exceeded the regular component.
Thus there is no physical basis for extending to the asteroids the function
k (m) obtained for the planets. A single power function was obtained by Hart-
mann and Larson at the expense of a considerable reduction in accuracy for
the planets. Thus with regard to the initial period of rotation of the Earth,
MacDonald's approximation is unquestionably the one to be preferred.
25
,20
/s
<*
\
R
o
\
>
°8
o
B
\
o
o
o
o
o <
o
o
o
52
C
6
'a.
...
o
Ast
1 1
o a
s r oi
1
ds
— -J
1 1
1 1
fi fS 20 2/ 22 23 2t 25 26 27 28 15
_L
' » ' ■
-J I l_l I
/3/ItftffSt 7 6 5 U 3 Z /
Abs. mae.
Mass
JO J/
log/37
FIGURE 7. Period of rotation of planets and asteroids as a function of their
masses. The straight line A passes through all planets besides the Earth
and Uranus. The straight line B (same angular velocity) passes through the
the three giant planets and asteroids with wide dispersion of points.
But MacDonald's approximation is not the only possible one. In Figure 7
the values of the period of rotation P r are plotted as a function of log m. The
points fit more comfortably on the straight line than they do in MacDonald's
graph. For bodies with the mass of the Earth, the dependence yields a
period of rotation of about 20hrs. Linear approximation on a log P r — log m
graph gives approximately the same value for the period, and the dispersal
of points is 1.5 times less than on the log k —log m graph. The divergence in
the values of P r for the Earth on different graphs can be explained with the
127
aid of the relation kcc pwr 2 cc pZ'^m^fP,. The planet's coefficient of inhomo-
geneity ^ is approximated by a power of m without substantial distortion.
But due to the differences in chemical composition between planets, their
densities differ widely independently of their masses. The presence of the
factor S~ Vs leads to various deviations in the actual values of P r and k from
the monotonic (power) function of m. Taking relation (2 7), which was
obtained from considerations relating to the asymmetry of impacts from
falling bodies, we obtain
p r oc iir* k oc r\
The Earth is denser than other planets from which the functions k (m) and
P r (m) were determined. Introduction of the appropriate correction increases
the value obtained by MacDonald for the initial period of rotation of the Earth
from 13.1 to 14.4 hrs and reduces the value obtained from the function P r (m)
from 20 to 15 hrs. These values of P r correspond to an initial distance
between Moon and Earth of about 45 Earth radii.
128
Chapter 11
THE INCLINATIONS OF THE AXES
OF ROTATION OF THE PLANETS
30. Evaluating the masses of the largest bodies falling on the
planets from the inclinations of the axes of rotation of the planets
One of the most serious difficulties encountered in the development of a
theory of planetary accumulation is the scarcity of observational data
capable of serving as checks on different parts of the theory. The data that
exist are limited chiefly to the laws of motion and planetary composition.
Any opportunity to make use of these data is very important for the theory.
It was discovered by the author (Safronov, 1960a) that the inclinations of the
planets' axes of rotation are related to the random character of the impacts
of individual bodies falling on the planets during the accumulation process,
and that the sizes of the largest bodies that fell on them can be evaluated
from these inclinations. An estimate performed with allowance for the size
distribution function of bodies (Safronov, 1965a) revealed that the masses of
the largest bodies settling on the Earth amounted to about 10" 3 times the
Earth's mass. From Section 26 it is evident that this mass ratio is related
by expression (9.7) to the bodies' relative velocities and that it makes it
possible to evaluate the parameter characterizing these velocities.
Knowledge of the sizes of the largest bodies that fell on the Earth is also
important for geophysics: it is required for the determination of the Earth's
initial temperature (see Chapter 15) and makes it possible to estimate the
scale of primary inhomogeneities of the Earth's mantle (see Chapter 16).
In Chapter 10 we remarked that the observed rotation of the planets
breaks down into two components: a systematic (regular) component with
momentum K t at right angles to the central plane of the planetary system
(direct rotation) and a random component K 2 manifested in the inclination of
the planets' axes of rotation. The latter is related to the discreteness of the
process of planetary growth. It shows that a considerable fraction of the
mass settled on the planet in the form of individual bodies with randomly
oriented relative motion at the instant of impact. A characteristic feature
of the planetary system is that the angles of inclination of the axes of most
of the planets are of the same order of magnitude. It points to a definite
pattern of growth, to a pattern governing the size distribution of the bodies.
Let m and r be the mass and radius respectively of a growing planet, and
let m\ be the mass of a body falling on it. For clarity we will begin with the
case where all falling bodies have the same masses m' t = m! and move in the
plane Oxy with reference to the planet m whose center lies at the point O. Let
v be the velocity of a body with reference to the planet before impact. Then
129
the angular momentum imparted to the planet by the mass m i}
is directed along the z-axis and is a random variable, since the impact
parameter I. of the incident body is a random variable with constant
probability density in the interval ( — Z , +Z ). The expectation value
of / (mean value of l) is zero, but the expectation value of I 2 (variance
of /) is nonzero:
— '•
The quantity l , the largest impact parameter leading to collision between to'
and to, is related to the radii r and r' by relation (9)
2G{m + m') -\ (3)
/y=(r + w)'[^+ 2G y/> ],
which is an elementary consequence of the laws of conservation of energy
and angular momentum in a two- body system.
For m'v=const, when several bodies to' fall on the body to we have, from
the theorem for the addition of variances of a sum of independent random
variables (Gnedenko, 1962),
d 2 AK *< = ( m W D 2 '< = ( m ' v ? 2 D/ < = < m ' y ) 2 ^ • < 4 )
Consequently, the mean value of the square of the angular momentum
imparted by bodies m* with total mass Ato=mto' is given by
*Kl = {m'u)*Q = (vljr%6m. (5)
The magnitude of the random component AK 2 of the angular momentum
imparted to the planet by incident bodies is obviously determined by its mean
square value, which is related to to' by expression (5). From (5) it is
obvious that the angular momentum imparted increases with the size of the
body to'. Small particles contribute practically nothing to AK t .
In the more general case of bodies moving in all possible directions the
random component of the imparted angular momentum can be evaluated as
follows. Let one third of all bodies (n/3) move parallel to the x-axis, one
third parallel to the y-axis, and one third parallel to the z-axis. This
simplification is frequently employed in the kinetic theory of gases.
Consider the bodies moving along the z-axis before encountering the
planet. When they fall onto the planet they will impart to it the angular
momentum components K Ux and K ttf along the x- and y-axes respectively.
Obviously,
K ux = m ' vl sin 9* R% iv = m'vl cos f i ( 6 )
where <p is the angle between the plane Oxz and the plane of the body's orbit
130
with reference to the planet. The variance of the random variable K tfx is
given by
/ 2x
j j {l sin <p)*ldld<p
J I ldld *
Similarly, DK Ulf = ( -^^
«Jf~ 4
An angular momentum component along the x-axis will also be contributed
by bodies moving parallel to the y-axis before encountering the planet; the
variance DK tix is given by the same expression (7). The variance of the sum
of random variables K 2{x is equal to the sum of the variances of the terms
*>'% K «' = 2 T DK "- = TW'*f- (8)
»=1
The angular momentum components along the y- and z-axes will have the
same variance. According to (8), the expectation value of the
square of the angular momentum component along the x-axis is
aa-2 — vn W^ (9)
UA fcr g .
Consequently,
A/q = W\ x + A* \ v + £Jd = -i- i*X*». ( 1 0)
We introduce &l\ from (3) above, taking i?=Gm/Qr in the right-hand side
in accordance with (7.12) ( 6 is of the order of a few units) and dropping the
terms m' and r' , which, as will be evident from what follows, are small
compared with m and r. Then
AK* = (l + 4) Gmrm'ton = (l + -gy) Cmmm*. (11)
The specific angular momentum imparted is inversely proportional to the
square root of n:
AKJAm = V(l + 1/29) Gmrln. ( lv )
From the rule of summation of variances it is easy to obtain an expres-
sion for Atff in the more general case where the masses m'j of the falling
bodies vary. To do this expression (11) must be summed over all m Jt Let
n (m') be the mass distribution of bodies incident on the planet and having
total mass
in,
Am = j win (m!) dm!. m 2 )
131
Integrating (11) over all m! and introducing Am from (12), we obtain
mi
I n (m')m' 2 dra'
A*J*»(l +^)Gmr^ r< Am, < 13 )
\ n («') m'dm'
o
where mj is the mass of the largest body, not counting the planet itself.
This relation is obviously meaningful only for m x <^ Am <^m.
The expression
♦»,
J n (m') m' 2 dm'
/(«. *,) = -==; (14)
m f n (m') m'dm'
is a function of the planet mass m, since n(m!) depends on time. If
n(m\ = c(0m rf , (15)
then for q < 2
?' m' 2 "W
i j" m fl ~ q dm'
v i; «• 3 — q m '
The masses of the falling bodies m! increase with the planet's growth,
and therefore mjm can be regarded as constant in the first approximation.
Then 7= const if q = const. Regarding the planet's density as constant and
integrating (13) over m, we obtain the square of the random component of
rotational momentum of the planet
and thus
K *= m VW L+ -&) JGmr - (17)
Allowance for increasing planet density with m scarcely affects the result;
the right-hand side of (17) increases only by a quantity of the order of 1% in
all.
By definition the vector K 2 has a random orientation in space. Let the
angle between the systematic angular momentum component K x perpendicular
to the orbital plane and K 2 be ip, and the angle between K 2 and the total
angular momentum vector of the planet K = Kj-f- K 2 (inclination of the axis of
rotation) be e. Then the angular momentum component perpendicular to z is
given by
K % sin <J» = K sin e. (18)
132
The right-hand side of the above is known from observations. In the left-
hand side K 2 is given by relation (17) and the angle \jj can take any value
between and it. For the probable value of sin0 in (18) it is natural to take
its mean value. For uniform distribution of the vectors K 2 over the sphere
sin 2 <|> = — \ sin ? ty2n sin tydty — -y .
(19)
Introducing sirrfy and K 2 expressed in terms of mjm with the aid of (16) and
(17) into (18), we obtain the expectation value of mjm:
m x 3 — q 5sin 2 e A' 2
~m~~~2^q {i + 1/26) Gmlr '
(20)
For numerical computations it is convenient to introduce the velocity of
rotation v r of the planet at the equator and the circular Kepler velocity v c
at the planet's surface:
K = -jr- pmrv r ,
Then from (20) we obtain
m l 3 — q
m ~~ 2 — q 5(1 + 1/26)
{,1 sine^) 2 .
(21)
(22)
The masses of the largest bodies falling on the planet as calculated from
this formula for a power distribution function with q = 5/3 (distribution over
radii with exponent p = 3g-2= 3) are given in the first column of Table 12
and in Figure 8.
FIGURE 8. Masses of largest bodies m x falling on planets during
their period of formation, evaluated from the inclination of the
planets' axes of rotation. The unit is the planet mass m.
Lower curve — all incident bodies were of the same size. Central
curve — incident bodies had power law of mass distribution with
exponent 7 = 1.5. Upper curve — inclination of axis of rotation
due to fall of a single body of mass m u .
For Uranus sine was replaced by the ratio kKJ^K, which was obtained
under the assumption that the systematic angular momentum component K x
of Uranus corresponds to a period of rotation of 15hrs (approximately the
same as for Neptune).
133
TABLE 12
mjm
Planet
Q = 5 A
q = —oo
m n fm
Earth
Mars •• .*
Jupiter
Saturn
Uranus
Neptune
1.10-3
2-10-3
3-10-4
4. 10-2
7-10-2
7 - 10-3
3.10-4
6 • 10-4
9 - 10-5
1 . 10-2
2 . 10-2
2-10-3
1-10-2
1.3- 10-2
5-10-3
6-10-2
8-10-2
2.10-2
For 0> 3 the role of the parameter characterizing the relative velocities
of bodies before approach to the planet in (22) is insignificant. We have
assumed 6 = 3. For q = 1.8 the values obtained for mjm are twice as large
as for q = 5/3.
For q = 2 (i.e., p = 4), (3-q)/(2-q) in the right-hand side of (22) should
be replaced by In (mjm^), where m m)n is the mass of the smallest particles
in the given distribution. Here the values of mjm become 2—3 times as
large as for q = 5/3. In Section 25 it was shown (see Chapter 8) that q lies
in the interval 3/2< q <2 and that it is probable that it does not depart
considerably from the value q = 1.8.
Computations were carried out under the assumption that mjm^ const.
Variation of mjm affects the results only weakly: for m x oc m % the values of
mjm would be 30% higher than given in Table 12 for mjm= const, while for
m x = const its values would be 30% lower.
The second column of the table lists the values of m x lm for q=— co, i. e.,
for the case where all falling bodies are of the same mass. These values
are three times smaller than the preceding set. One could consider yet
another extreme case where the random angular momentum component K % is
imparted only by a single body m n while all the remaining matter falling on
the planet imparts to it only a regular rotation (K x ). Then
K sin e = i K 2 = \m n Jv = ~ m n \ l v
(23)
and the expectation value of m n /m is given by
= — ■ - — - sn
m 5Wl -f- 1/20 f*
(24)
The values of mjm are given in the last column of the table. They vary less
from planet to planet than do the values of m x im. The values of m n /m can be
regarded as the upper limit for the masses of bodies falling on the planet.
We see from these results that despite the absence of definitive data on
the size distribution function for the bodies, the masses of the largest bodies
falling on the planet in the process of their formation can be determined with
relative certainty, with no more than threefold deviation in either direction.
The 3 masses of the largest bodies incident on the Earth amounted to about
10 Earth masses. Due to the tidal effect of the Moon the Earth's rotation
is slowing down, and although the axial inclination e is increasing, the
134
quantity v r sine is decreasing (Gerstenkorn, 1955). If one assumes that the
Moon's original distance was half the present one, the value of mjm obtained
above for the Earth should be increased slightly less than twice.
The retrograde rotation of Uranus can be explained naturally by the relatively
larger sizes of the bodies from which it was formed. The masses of the
largest bodies falling on Uranus reached 0.07 planet masses. The bodies in
Saturn's zone of formation were also of considerable size. The largest of
these amounted to 0.04 planet masses. Consequently, with regard to rota-
tional anomaly, Saturn differs only slightly from Uranus. The causes of the
anomalies are related to the influx into the zones of these planets of larger
bodies from the Jupiter zone. In Section 31 it will be shown that Jupiter
grew much more rapidly and that it began scattering bodies earlier by virtue
of its own gravitational disturbances into the zones of other planets. It
should be mentioned that the estimates of mjm cited above for Jupiter and
Saturn require substantial revision as they fail to account for the accretion
of gaseous hydrogen in the closing phases of growth of these planets.
However, such revision will be possible only when a theory of growth has
been developed for these planets.
135
Chapter 12
GROWTH OF THE GIANT PLANETS
31. Duration of growth process among the giant planets
The growth of the giant planets was complicated by a number of important
factors, including first and foremost fusion of source zones, ejection of
bodies beyond the solar system by gravitational disturbances, dissipation of
gas away from the giant planet region, and the accretion of hydrogen by
Jupiter and Saturn.
Evaluation of the planets' rate of growth indicates that for the outermost
planets (Uranus and Neptune) formulas like (9.18) yield an inadmissibly long
growth span — 10 11 years (Safronov, 1954). To circumvent this difficulty one
would have to assume either considerably lower relative velocities for the
bodies or a considerably larger mass of material inside this region
(Safronov, 1958a).
Table 13 gives the characteristic time t of exhaustion of available matter
by the giant planets, as calculated from formula (9.16) for their present
masses and densities under the assumption that the planetary zones were
isolated. It would appear from the table that the distant planets (Uranus,
Neptune and Pluto) could not have managed to develop and use up all the
matter in their zones within the lifetime of the solar system. For Neptune
i is 10 2 times greater than the maximum admissible value.
TABLE 13
Jupiter
Saturn
Uranus
Neptune
0.25
3 5
148 94
0.001 0.002
0.23 0.50
Pluto
o
95
3 5
0.055 0.035
0.12 0.26
0.70 1.5
1.30 2.7
1.6 3.5
3.2 6.8
5.7
3 5
1.02 0.65
0.07 0.15
0.23 0.50
0.35 0.75
1.3 2.8
0.3
3 5
47 30
0,009 0.02
0,03 0.06
0.45 0.99
10" 3
a
3 5
417 265
x , billion years . - - .
m 8 (Earth masses) • .
m
N
0.15 0.33
In the initial stage of growth, when the masses of these planets were still
small, their zones did not overlap. The growth formula (9.18) enables us to
obtain the growth times of the planet embryos at this stage if the initial
surface density a of matter in their zones is known. Some idea of the
136
duration of the initial stage of growth can be obtained by examining the next
rows of the table. They give the planet masses for which bodies having the
corresponding relative velocities (v = \ / Gmfir, for 6 = 3 and 6=5) will be at
aphelion along their orbits at the distances of the other planets (indicated by
the subscripts on m) if v is directed along the orbit and forward (with refer-
ence to the planet's motion). The masses are given in terms of the Earth's
mass. Thus the mass of Jupiter for which bodies could have traveled from
its zone to the distance of Saturn is given in the column headed Jupiter and in
the row designated by m a . For 6=3 and 5 it is 0.12 and 0.26 Earth masses.
Since the inner planets grew more rapidly than the outer ones, the zones of
Jupiter and Saturn were the first to fuse. At this stage m<^Q, and one can
use the simplified growth formula (9.21). If the accretion of hydrogen by
Jupiter had not yet begun at this time, the a appearing in t should refer to
solid matter alone. The mass taken in Section 32 for the protoplanetary
cloud (0.05 Af ) corresponds to a «20g/cm . Bodies from the Jupiter zone
should then have flown out to the distance of Saturn within 50—60 million
years after Jupiter had first begun to form, and to the distance of Neptune
within 100—130 million years; they would have escaped beyond the solar
system within 150—170 million years. The escape of bodies from Saturn's
zone began considerably later — 10 9 years to the distance of Uranus.
Thus already 150 million years after the large planets had begun to grow,
bodies from the zone of Jupiter were shooting through the entire outer portion
of the solar system as well as through the zones of the asteroids and Mars.
Since the zones of these planets ceased to be isolated, the foregoing formula
of planetary growth becomes inapplicable. The growth process was further
complicated by the presence of gases. At first the role of the gases was
confined to lowering the relative velocities of particles and bodies by retar-
dation. As we saw in Section 22, the value of may have been substantially
higher. Since the masses of planetary embryos permitting escape of bodies
(given in Table 13) are proportional to 8'' 1 and the rate of growth is propor-
tional to 1+ 29, in the case of Jupiter (for example) for 6= 30, m 8 = 3.9 and
the time required for growth to this mass is about 60 million years. For
this value of the mass accretion of hydrogen will already have taken place,
leading to considerably faster growth of the planetary embryo due to the fact
that the radius r a of capture by accretion is proportional not to the radius of
the embryo but to its mass:
:>/S = -
Gm
l?2 + C 2 »
where 1< a< 2; v is the velocity of the gas with reference to the planet
embryo and c is the thermal velocity of the molecules (Bondi and Hoyle,
1944; Bondi, 1952).
When the mass of the embryo is sufficiently large, the energy imparted
to it by the gas leads to considerable warming of its surface. An approxi-
mate computation for i?*-f-c* = 1 km/sec (Safronov, 1954) indicated that the
maximum temperature reached for mass equal to 2/3 of the contemporary
planet mass was 17,000° for Jupiter and 3600° for Saturn. The result
depends on the estimated gas density, which is unreliable in view of the fact
that the initial mass of the cloud and the rate of dissipation of the gas are
unknown. But since the temperature is determined from the balance of
absorbed and emitted energy and is proportional to the fourth root of the gas
137
density, the estimate should be correct as to order of magnitude. The
higher densities of satellites adjacent to Jupiter are probably attributable
to the high temperature of Jupiter during their formation.
The growth times of Uranus and Neptune can be determined approximately
from expression (9.21) for the growth of bodies in a medium of constant den-
sity, which can be written as follows:
Ptr „P8r ( 2 )
(1 +20)a — 28a *
From here it is easy to find the value of o6 required for the planet to
complete its growth within a time not exceeding 4 billion years. Then for
Uranus ^6 > 40 and for Neptune oG > 80. Since a<a > one needs o 6»10 2 . The
high values of a 6 in the outer- planet region lead one naturally to conclude
that the ejection of bodies from the solar system played a considerable part
in the process of formation of these planets.
32. Ejection of bodies from the solar system
The ejection of matter from the solar system is already mentioned by
Oort (1950, 1951) in his theory of the origin of comets. Oort conjectured
that the comets were formed together with the planets in a single process,
and were ejected by Jupiter's perturbations from the asteroid zone beyond
the confines of the solar system. About 5% of the total number of bodies
ejected continued to travel around the Sun at large distances under the
influence of perturbations from' stars closest to the Sun within the so-called
comet cloud. There they were effectively preserved owing to the low tem-
perature. They occasionally reenter the planetary system under the influ-
ence of renewed stellar perturbations, becoming observable as comets as
they draw near the Sun. Levin (1960), having noticed that in the closing
phases of growth the relative velocities of bodies in the giant planet region
(which are proportional to the parabolic velocity at the surface of the planet)
exceeded the parabolic velocity at the distances of these planets from the
Sun, concluded that the ejected mass may have been substantial and that the
masses of the giant planets do not determine the initial mass of matter in
this region; they represent instead a kind of limiting mass. Having achieved
this mass the planet practically ceases to grow further, since in the main it
ejects the bodies drawing close to it and fails to use them up.
If the mass of solid matter ejected was considerable, the total initial
mass of the protoplanetary cloud must have been correspondingly greater.
The lower limit of the cloud mass is usually evaluated by adding volatile
substances to the matter of the planets until the solar composition is reached.
According to Whipple (1964), to reach the solar composition the mass of the
planets in the Earth group must be increased 500 times; the mass of Jupiter
must be increased 10 times, that of Saturn 30 times, that of Uranus and
Neptune 75 times. This gives a minimum initial mass for the cloud of
0.028 Mq.
Whipple gives 0.003 Af Q for the mass of ejected substance and 0.031A/ o
for the total mass of the cloud. Data for Jupiter and Saturn are not in con-
tradiction with computations of the hydrogen content in these planets carried
138
out by Kozlovskaya (1956). However, these results were based on models
of the giant planets not containing helium. The addition of helium reduces
the content of heavy elements and brings the composition closer to the solar
one. Kieffer (1967) has recently constructed a model of Jupiter using
material of solar composition and one of Saturn using material roughly 5%
more dense. The lower limit of the cloud mass for this particular compo-
sition of these planets decreases by a factor of 2—3. However, the
estimated compositions -of Jupiter and Saturn remain unreliable in view of
the absence of a reliable equation of state for hydrogen and helium mixtures.
A "ceiling" estimate of the initial mass of the cloud can be carried out
independently for the gaseous component and for the solid matter (Safronov,
1966a).
1. All the planets (with the exception of Jupiter and Saturn) differ substan-
tially in chemical composition from the Sun, and none could have formed as
a result of gravitational instability in the gaseous component of a cloud whose
composition is assumed to be solar. The mechanism of thermal dissipation
could not have produced practically total sorting and removal of hydrogen
and helium (originally amounting to 987o by mass) from massive self-
gravitating condensations (Shklovskii, 1951). With formulas (6.6) and (5.17),
it can be shown that condensations, formed in the giant planet region as a
result of gravitational instability inside the gases, should have had masses
equaling about 15 Earth masses. It seems one must rule out the possibility
of hydrogen and helium separating out of such massive bodies under the
conditions prevailing in the protoplanetary cloud.
It cannot be concluded from the chemical composition of Jupiter and
Saturn that gravitational instability could not have been maintained in the
gases of their zones. However, the assumption of such instability raises
considerable difficulties. For instability to have appeared in the gas the
gas density must not have fallen below the critical value 2.1 p*(see Section 16)
and the total mass must not have been less than 6 Jovian masses in each
zone. Instability could have spread over the entire planetary zone, leading
to the formation of about 10 3 condensations. If, on the other hand, instability
affected a small part of the zone, the number of condensations may have been
small. However, in neither case is it clear why the process of interaction
and fusion of the condensations into a developing planet involved only 1% of
the matter inside the zone while 99% was ejected from the solar system.
Below it will be shown that the mass of bodies ejected by the planet in
encounters between it and the bodies is merely one order greater than the
mass of bodies falling on the planet. This could induce one to believe that
gravitational instability was absent in the gases of the Jupiter and Saturn
zones.
Thus the gas density p in the central plane of the cloud need not have
reached the critical density 2.1 p*. A reasonable upper limit for p could be
the value p*, corresponding to the total mass of the cloud in the zone of the
large planets, namely 0.12 M Q . It has also been mentioned that it is neces-
sary to bound the cloud mass from the top by the value ~ 0. 1 M 0J which is
determined by the possibility of dissipation of this entire mass of gas from
the solar system (Kuiper, 1953; Ruskol, 196 0).
2. A large initial cloud mass implies dissipation of a large quantity of
solid (not just gaseous) material from the solar system. The mechanism of
ejection of bodies by gravitational perturbations appears to be efficient, but
it entails grave consequences related to the redistribution of angular
139
momentum. An ejected body must raise its absolute velocity to the para-
bolic velocity. When a body encounters the planet, its relative velocity
vector rotates without changing magnitude. Its absolute velocity will
increase if this rotation takes place along the direction of orbital motion
of the planet (whose orbit can be regarded as circular). In the process the
angular momentum of the body with reference to the Sun will increase due
to the orbital momentum of the planet. Consequently bodies are predomi-
nantly ejected in the direction of the planet's motion. If the total mass of
the ejected bodies is comparable to the mass of the planet, the planet will
draw noticeably closer to the Sun. This might account for Neptune's
violation of Bode's law: for the distance from Neptune to the Sun to decrease
from 40 to the present-day 30 a. u., it is sufficient that Neptune should have
ejected from the solar system a mass of bodies equal to one third of its own
mass.
The condition of angular momentum conservation gives the relation be-
tween the mass dm e j ejected from the solar system from a distance R in the
direction of rotation and the Sunward displacement -dR of the remaining
mass m :
(yjT — \)slGM Q R dm e ^ — md\jGM Q R. (3)
Ejection of the mass m ej -causes the distance R of the mass m from the Sun
to reduce to R' } as given by
m e j=l-211n(-^-)m. ( 4 )
For /*//?'=*/>, m e j^m/3 and for R!R'=*/ tt m ei -^m/2. Assuming that the mass
of solid material in the giant planets equaled 50 Earth masses and that the
mass of solid material in a cloud of solar composition would equal 1/75 of
the cloud mass, we find that in the first case (R/R'= l / 3 ) the mass of the
protoplanetary cloud is 0.05 M e and in the second case it is 0.06M©. The
larger values of R/R' seem implausible since the small mass of Mars and
the arrestation of the process of fusion in the asteroid zone show that
Jupiter's distance from the Sun during its formative period did not substan-
tially exceed its present distance. It would also be difficult to accept the
larger values of m t j, since the amount of solid material ejected by the
planets should not have been far in excess of the amount included in the
planets. Therefore 0.05 — 0.06 M Q seems to us a reasonable upper limit
for the initial mass of the protoplanetary cloud.
When the Jovian embryo had grown to the size of 2—3 Earth masses it
scattered a considerable portion of the material in its zone over the entire
large planet region. The addition of 100—200 Earth masses in the Uranus
and Neptune zones would make it possible to raise the surface density O of
solid matter in these zones by one order. For Neptune to grow at the
required rate, one must further increase the efficiency of collisions between
the bodies and Neptune by one order, i. e,, increase the gravitational focus-
ing while reducing the bodies' relative velocities (increase G). At the initial
stage of growth may have been appreciably larger owing to damping of the
bodies by the gases (Section 22, Table 7). But by the time Neptune reached
a mass of terrestrial order the gases had largely escaped from its zone:
otherwise there would have been accretion of gas by the planet (see Section 33)
140
and Neptune would have contained appreciable amounts of free hydrogen. By
the time Neptune reached a mass such that it could eject bodies in its zone
beyond the confines of the solar system, relative velocities of bodies in its
zone practically ceased to increase further, since all bodies whose absolute
velocities V grew to the parabolic velocity V, left the system. If we desig-
nate by y e j the mean relative velocity at which ejection took place, we have
y ej — VGm/6 ej r . Further increase in the planet mass was therefore accompanied
by an increase in 6 ej -.
Ejection of bodies proceeded predominantly in the direction of rotation of
the system. The velocity vector v e j of the ejected body deviated from the
tangent to the circular planet orbit by an angle not exceeding <p, related to
by
whence
* rl = * r ; + £ j +2F^ j cos* = F2 = 2F;,
fej= V t (VI + cos 2 9 — cos f ) = V € u («p).
The angle <p was determined by the efficiency of the mechanism of growth of
the bodies' relative velocities. It increased as the planet mass grew,
amounting to about 45° in the last stage. Table 14 lists values of v e j/V e as
a function of cp, as well as corresponding values of ej computed for present-
day masses and radii of the giant planets and typical, therefore, of the
concluding stage of growth.
TABLE 14
<P, '
15
30
45
60
'ejlV. ■ ■ ■
*J
•«
*u
"*
0.414
63
40
31
60
0.424
60
38
30
58
0.456
52
33
25
49
0.516
41
26
20
39
0.62
28
18
14
27
The parameter 6 characterizing the mean relative velocity of all bodies
in the planetary zone is obviously greater than 6 eJ *:
In the closing stages of growth of the planetary giants for a Maxwellian
velocity distribution 6«(2— 3) 6 e j.
Thus does appear to have been of the required order of magnitude in the
last stage of growth. But it is unclear what 6 was in the intermediate stage.
Bodies collided infrequently owing to the low density, and "foreign" bodies
(impinging from other zones) had higher velocities. No one has analyzed
141
the accumulation process in this region while allowing for all essential
factors, and as yet there is no theory of growth of the giant planets.
It is still not clear how much solid material was ejected from the solar
system in the course of the giant planets' growth. Oort and Whipple suppose
that its mass was one order greater than the present mass of the comet cloud,
i. e., about 1-10 Earth masses. The inference that the ejected mass was
small is drawn in an interesting work by Opik (1965) devoted to dynamical
aspects of comet formation. It is based on an analysis of the mechanism by
which bodies speed up during encounters with a planet. Opik concludes that
only large bodies of the order of comet nuclei (1 to 100km in diameter) could
have been ejected over large distances. Smaller bodies were braked in col-
lisions with other bodies and remained inside the system. But Opik presup-
poses an excessively high density of matter in the system — p a 10" /, where
/ is the fraction of the planet mass that had not been used up. If the fraction
; of the present mass of Jupiter were to be scattered over the entire large
planet zone up to the distance of Neptune, the mean density of matter in this
volume (equal to 0.4—0.5 for contraction along z) would be 10" /', i. e., four
orders of magnitude smaller. The lower limit obtained by Opik for the sizes
of the ejected bodies must therefore be decreased from 1km to 10 cm. More-
over, Opik assumes that bodies encountering the planet increase their
relative (random) velocities only in the case of elliptical planetary orbits.
For small eccentricities of the planetary orbit the velocity dispersion of the
bodies increased slowly. This apparently holds when the variation of the
bodies' orbits is affected by perturbations from one planet alone. Owing to
the low density encounters among bodies were rare, and the resulting mutual
perturbations small (although they should have been substantial for the
density assumed by Opik). But the proximity of other planets or their
embryos (see Section 26) would cause the velocity dispersion of the bodies
to increase even if planetary orbits were circular (the mechanism of velocity
growth being similar to that described at the end of Section 23).
It follows from these remarks that the mass ejected by the planets could
have been appreciably larger than Opik suggests. We saw earlier that unless
we increase the mass of the ejected material accordingly, the growth times
of the distant planets would involve us in considerable difficulties.
One can make a rough estimate of the amount of material ejected using
the results of Chapter 7. Encounters between bodies and a planet are char-
acterized by the relaxation time T m D and encounters among bodies themselves
by the relaxation time T" p . Within the time x # , the inverse of which is equal
to the sum of the inverses of T), and T% (see 7.84), the body's relative velocity
vector v turns as a result of encounters through an angle w/2 on the average.
Within the time x, # (which differs from % in that, if rj< Ti , it will contain
T n D instead of Tl) the rotation of v causes bodies to acquire an energy of
relative motion amounting on the average to the following quantity per unit
mass (according to (7.29)):
V*i = P V - (6)
At the closing stage of growth t„ > t, and the number of turns of the vector v
involving an increase in v is substantially less than the total number of turns.
The turns are mainly due to distant encounters, which are one order more
efficient than close encounters. Body velocities thus increase gradually to
(see (5)). Further growth ceases since bodies reaching this
142
velocity leave the solar system as soon as the vector v enters a cone of
aperture <p and axis coincident with the direction of motion of the planet
around the Sun.
Let dfi e j De tne fraction of bodies ejected within time dt. The equation
of conservation of energy per unit mass can be written as
«fc i + (^ j -^)^ej=2(t I -i 1 )d* l (7)
where v*. is the mean square relative velocity of the ejected bodies and
e 2 = C ; t? 2 /^ is the energy of relative motion lost in collisions among the bodies.
We denote by X the ratio of the frequency of ejections of bodies from the
solar system to the frequency of incidences on the planet. Then the lifetime
of a body before ejection is given by x*Jl, where ?] is the lifetime before in-
cidence on the planet. Since
, 2 i/Gm\ (1dm db \ 2 2 2 0— m, 9 d9
we have
. 2(pv^Hi-. s / El )d/ + d e/8
ct P*ei == — 5 ■ v • (o)
rej vl r v2[i-2(Q-m)/3m\] V '
In the closing stage of giant planet growth X > 1 and 2 ( Q—m)/3mX < 1. The
second term in the numerator (46/6) is also small and can be disregarded.
Moreover, the expended energy e a is several times smaller than e x . The
fraction of bodies landing on the planet in time dt is given by dtjz].
Therefore
ej »*
Assuming Tl = xT*> X > 1 and * 9 , = x t dI2, we obtain, from (7.81) and (7.82),
x<r; i6P7 8 i>» eg _ 4p»/ 5 v* m )
x(<>* r » 2 ) *+»^ x pj r p«'
where i>, = y/2Gmjr = y ^26" is the parabolic velocity at the planet surface.
According to (7.79), / 3 «? 2 In (D M fir) -1. The expressions for the relaxation
times were obtained from computations of the angle of deflection in encoun-
ters carried out within the two-body problem. Therefore the maximum
distance D M to which the overall result of distant encounters is reckoned
should not exceed the radius of the sphere of action or the distance to the
libration point L x of the planet m. For Jupiter r z ~10 3 . Then / 3 ^4. The
mean square relative velocity v 2 increases with time; for a Maxwellian
velocity distribution its upper limit is ( 9 / 5 )w*-- One can take v 2 mvl } /3. From
Table 14 it is clear that for <p <6 0° the ejection rate is close to VJ2. Lastly,
from the results in Section 21 we can take p"«0.13. Consequently,
X<!2-£ (11)
^x v y
For Jupiter we obtain Xj^200/x. The parameter x introduces the largest
error in the X estimates. According to (7.87), if in addition to the planet m
143
there were n bodies of total mass vm, then
If n =10 2 and v = l, then X^ 15; if «= 2 and v =1/5, then x»12. In Jupiter's
last stage of growth it is probable that x > 10. For x = 15 we obtain X= 15.
If the composition of Jupiter is similar to the Sun's and the amount of
material it contains in solid form amounts to about 10 Earth masses, then
for this value of A the amount of solid matter ejected by Jupiter is equal to
half its mass. This estimate agrees with the one obtained earlier from
expression (3), which was based on analysis of the angular momentum lost
by the planet upon ejection. It is nonetheless very approximate and as yet
no final conclusions may be drawn from it.
33. Dissipation of gases from the solar system
Dissipation of gas from the solar system took place in parallel with the
evolution of the dust component of the protoplanetary cloud. Since the Earth
was growing for ~10 years, the absence on Earth of significant amounts of
hydrogen and helium as well as the striking deficit in the noble gases (Brown,
1949) lead one to suppose that these gases escaped from the region of Earth
group planets within a period not exceeding 10 75 years. Uranus and Neptune
also lack significant amounts of helium and free hydrogen. By the time they
had become sufficiently massive, the gases had already escaped from their
region. The dissipation time probably did not exceed 10 years. Jupiter
and Saturn developed a good deal earlier than Uranus and Neptune. They
absorbed all the gas that had not yet escaped from their zones.
It is sometimes assumed that accretion will begin when the body mass is
such that the capture radius for accretion exceeds the geometric radius of
the body. For a molecular thermal velocity of 1 km/sec this gives m ^ 10 g.
But bodies of such small mass are unable to retain a hydrogen atmosphere.
Accretion theory (Bondi, 1952), which is used to evaluate the rate of gas
absorption by gravitating bodies, does not allow for the reflected wave
which results when the falling gas strikes against the surface of a body.
Thus it cannot tell us for what body mass accretion becomes efficient. If
one calculates the atmospheric density for which the quantity of gas acquired
by a body by accretion will equal the quantity lost due to thermal dissipation
as computed by Jeans' well-known formula, it is found that the mass of this
atmosphere will cease to be negligible compared with the body mass only for
a body mass of about one to two Earth masses. The result depends on the
density and temperature of the gas. In the Saturn zone a body with this mass
could have developed in 500—800 million years. The gas should therefore
have been preserved in this zone for about 10 years.
Kuiper (1953) found that the most efficient mechanism of gas dissipation
from the solar system involves knocking out of atoms and molecules at large
z-values (in the rarefied portion of the cloud) by high- energy solar corpuscles.
The corpuscular flux would have to be large for a considerable mass of gas
to dissipate, but it was probably sufficient to ensure dissipation from the
region of the Earth group planets. Hoyle (1960) attributes the escape of
144
gases from the Uranus and Neptune region to thermal dissipation. At the
distance of Uranus a particle would dissipate in the direction of the cloud's
rotation at a velocity of 3 km/sec with respect to the circular Kepler veloci-
ty. Therefore in the Uranus region a gas temperature of 75° is enough to
ensure efficient dissipation of hydrogen and a temperature of about 150° K
for efficient dissipation of helium. In the presence of a dust layer such
temperatures would scarcely be attainable (see Chapter 4) due to rapid
cooling of the gas on solid particles. Gold (1963) regards the question of
gas dissipation as one of the most difficult ones in planetary cosmogony.
In his view the mechanism of thermal dissipation, like every other mechanism
involving uniform distribution of energy throughout the cloud, is inefficient.
Mechanisms providing concentration of energy in small volumes (solar flares,
corpuscular fluxes, perhaps even external interstellar "winds") require
appreciably less energy for the dissipation of the same amount of gas.
Schatzman (1967) found that at a distance of 2 a. u. from the Sun, dissipa-
tion of the gaseous component of the cloud would have taken place within
acceptable time limits if 4% of the energy of the Sun's emission had been
ejected during solar flares in the form of high- energy particles and ultra-
violet.
In our opinion the most convincing argument against thermal dissipation
of considerable masses of gas from the solar system is obtained by analyzing
the redistribution of angular momentum (as we did above to evaluate the
upper limit on the mass of ejected solid material). Thermal dissipation of
gas, like the ejection of solid bodies, takes place preferentially in the direc-
tion of rotation of the solar system. The remaining gas loses angular
momentum, shifts closer to the Sun, and should be absorbed by Saturn and
Jupiter. Thus in the absence of other mechanisms of dissipation, thermal
dissipation can account for the loss of only a small mass of gas not exceeding
the mass of Jupiter and Saturn, i. e., ~10 M Q) or 2—3% of the required
amount. Hoyle assumes a very small cloud mass (0.O1 M Q ). Still, this is
seven times greater than the mass of the planets. Expression (4) makes it
possible to determine the largest mass of gas which the planets could have
ejected. Since the actual dissipation of molecules must have taken place at
a certain angle <p.to the orbit (rather than precisely along it), the factor
u(<p) coscp should appear in the left-hand side of (3) instead of (VI— 1). But
this gives only a slight increase in the numerical coefficient on the right-
hand side in (4). For <p= 45° it equals 1.37 and for <p= 60°, 1.55. We then
find from expression (4) that even if the initial distance of the planets was
5 times greater than at present, the mass of gas they ejected could not have
exceeded the mass of the planets by more than 2—2.4 times. For a cloud
mass of 0.05 M Q the contradiction is even more striking. It is proof of the
ineffectiveness of thermal dissipation of gas from the solar system.
145
Chapter 13
FORMATION OF THE ASTEROIDS
34. Role of Jupiter in the formation of the asteroid belt
doers' theory of the formation of the asteroids due to the disintegration
of a planet (Phaeton), long popular among astronomers, has been rejected
by specialists in recent years. First, the possibility of a planet disintegrat-
ing is being seriously disputed from the standpoint of the physics of the
disintegration process itself. Second, it has been established that the disin-
tegration of a single planet would not account for the observed distribution of
the asteroid orbits, which points to a division of the asteroid system into a
series of individual groups (Putilin, 1953; Sultanov, 1953). Third, the
meteorites, themselves the products of the fragmentation of asteroids, also
fall into groups according to chemical composition (Urey and Craig, 1953;
Yavnel\ 1956). Fesenkov (1956) notes that the meteorites were formed from
bodies of asteroidal size which never combined to form a planet like the
Earth, since the crystalline structure characteristic of most meteorites
could not have been preserved at great depths in such a planet. Urey (1956,
1958) concluded from physicochemical studies of meteorites that at the time
of the meteorites' formation the material from which the planets developed
was in the form of solid bodies of asteroidal size." Shmidt (1954, 1957)
stressed that once it has been proved that bodies of asteroidal size could
have developed in the course of the planets' growth, there is no further need
for special theories regarding the origin of the asteroids. In the asteroid
belt the process of planet formation came to a standstill at the intermediate
stage of smaller bodies due to the proximity of massive Jupiter, which
increased their relative velocities. According to Shmidt, the asteroids'
position at the boundary between two groups of planets helped to slow down
their growth. Volatile substances originally present in the asteroids sub-
sequently evaporated, reducing their stability and leading to disintegration.
In our view the first argument can be accepted in full, the second only in
part (see below).
The mean eccentricities and inclinations of the asteroid orbits are «»0.12
and J«12° (Putilin, 1953; Piotrowsky, 1953). This corresponds to a relative
velocity (relative to the circular Kepler velocity) of 5 km/sec. This velocity
dispersion could not have resulted from interaction among present asteroids.
It follows from the expression v = \JGmfir , discussed in detail in Chapter 7,
that relative velocities of 5 km/sec associated with gravitational interaction
among bodies in a rotating system are possible only for body masses of the
* A detailed critical review of present-day hypotheses regarding the origin of meteorites was published by
Levin (1965).
146
order of the Earth's mass. It is probable that the observed velocity disper-
sion of the main asteroid mass is not another result of gravitational pertur-
bations emanating from Jupiter and accumulating throughout the lifetime of
the asteroids. A manifestation of these perturbations is the presence of gaps
in the region of values of the asteroids' periods of revolution commensurate
with Jupiter's period of revolution. These gaps, however, are very narrow,
and it seems they are due exclusively to limited variation of the semimajor
axis owing to perturbations from Jupiter (and thus to limited variation of the
eccentricity). Unfortunately expression (7.12) for v, which is based on
analysis of encounters and collisions of random character, cannot be used to
evaluate the cumulative effect of the interaction of two bodies moving along
nearly circular coplanar orbits.
A more probable conclusion is that the asteroids' velocity dispersion dates
back to the era of their formation. Earlier we saw that when the Jovian em-
bryo had become sufficiently massive the bodies in its zone acquired consi-
derable relative velocities and began to scatter into adjacent zones. Upon
colliding with bodies in the asteroid zones, they "swept away" most of these,
increasing the relative velocities of bodies remaining in the zone. In Section
26 we found that in the zone of action of the largest body the growth of other
bodies slows down due to reduced gravitational focusing. The bodies in the
asteroid zone were in a similar situation. At first their growth was slowed
down, eventually coming to a standstill when the energy of random motion of
the bodies had become substantially greater than the potential energy at their
surface and collisions among bodies began to lead to fragmentation rather
than fusion. Before this stage was reached both fusion and fragmentation
took place depending on collision conditions.
At present the kinetic energy of asteroidal bodies exceeds the potential
energy at the surface of the largest asteroids by more than one order, and
collisions between asteroids end in fragmentation. For a total mass of
material in the asteroidal zone amounting to about 10 3 terrestrial masses,
in 10 9 years every asteroid should experience collisions with other bodies of
total mass averaging about 10~ 2 of its own mass. For impacts at a speed of
several kilometers per second, this is sufficient to cause substantial des-
truction of the asteroids.
Thus the fusion of the asteroidal bodies into a single planet was hampered
by the proximity of Jupiter's massive embryo, which grew at an appreciably
faster rate. If the density of the dust layer had varied smoothly with increas-
ing distance from the Sun, the process could not have displayed such severe
irregularity. The asteroid belt lies at the boundary of the region of terres-
trial planets, however, and the reason for the anomaly is also the reason why
the planets fall into two groups. Condensation of the most abundant volatile
elements (CH4, NH3 and others) began at Jupiter's distance from the Sun
(Levin, 1949). This is evident from temperature conditions in the dust layer
(see Chapter 4). The surface density o of material passing into the solid
state was therefore several times higher in the Jovian zone than in the adja-
cent asteroidal zone. The critical density is proportional to p* and decreases
away from the Sun as R~ 3 . Therefore from the standpoint of the development
of gravitational instability in the dust component of the cloud, conditions
were far more favorable in the Jupiter zone than in the asteroid zone. For
instability to develop in the asteroidal zone, particle velocities, according to
(3.31), would have to be more than one order lower than in the Jovian zone;
147
similarly, the uniform thickness of the dust layer would have to be 1.5 orders
smaller. Thus there may have been no gravitational instability at all in the
asteroid zone, with simple growth of particles prevailing. If instability did
develop nonetheless, it must have led to the formation of condensations of
apDreciably lower mass. According to (6,6), condensation masses are pro-
portional to a 3 /? 6 . Condensation masses in the asteroid zone must therefore
have been two to three orders smaller than those in Jupiter's zone. Either
way, Jupiter's embryo was much larger than its neighbors in the asteroid
zone from the very beginning (Safronov, 1966b).
The growth process had a different outcome on the other side of Jupiter.
In Saturn's zone the surface density was nearly the same, and initial conden-
sation masses probably greater, near Jupiter. Jupiter's embryo grew more
rapidly (dm/dtcc o(^)i?" ,/j ) and gradually outdistanced Saturn's embryo. But by
that time the latter had become fairly large and the influx of bodies from the
Jupiter zone did not present a threat to it. In Uranus' zone the surface
density was lower and it grew more slowly. However, due to the greater R,
condensations masses were greater there. The influx of bodies from the
zones of Jupiter and Saturn did not interrupt the accumulation process,
although it did entail considerable disruption. The anomalous inclination of
Uranus' axis is likely to be due to impacts from these massive bodies.
Such in general terms are the features which characterized the process
of planetary growth near the most massive bodies — Jupiter and Saturn —
and which were responsible for the formation of the asteroid zone, the small
mass of Mars, and the anomalous inclination of Uranus' axis of rotation.
Qualitatively these features can be fully explained by means of the theory of
planetary accumulation developed above. However, to check these arguments
it would be necessary to study the accumulation process in this zone in
greater detail, bringing in all available observational data. The asteroid
belt is of great interest for cosmogony. To a large extent it preserves the
features of the protoplanetary cluster of bodies inside which the planets
were formed. The fragmentation products of the asteroids — meteorites —
land on the Earth where they can be subjected to a variety of laboratory
studies, making it possible to determine the physicochemical conditions
under which these bodies formed and evolved. Comprehensive study of
asteroids and meteorites thus constitutes one of the paramount tasks of
planetary cosmogony (Fesenkov, 1965).
35, Rabe's theory of the formation of rapidly rotating asteroids
Rabe (196 0) has suggested that rapidly rotating asteroids originated in the
fusion of asteroid pairs revolving around their center of gravity. He assumes
that the asteroids grew by gradually using up the finely- dispersed matter of
the protoplanetary cloud. The substance which acted as a feeding medium
for the asteroid embryo simultaneously served as a resisting medium. The
continuous growth of the masses of the asteroid bodies in the nutrient medium
and their retardation by this medium in encounters between single bodies, in
Rabe's view, could have led to the formation of pairs. The initially broad,
unstable pairs gradually became stable. In order for two asteroids of a pair
to converge from an initial orbit relative to the center of gravity of semiaxis
148
360 r to complete fusion, it is necessary that the radius r of each asteroid
increase 3.7 times (for a density of 2.0g/cm 3 ). When two asteroids combine
to form a single one, the result should be a body of elongated shape rotating
at the limit of rotational stability with a period of about 5hrs, which corre-
sponds to the observed velocities of rotation of the asteroids. Rabe believes
that pair formation in the present-day asteroid belt is nearly impossible,
since growth of bodies has practically ceased there, but in the past, in his
opinion, this process must have played an important role.
We will show that, despite the theoretical plausibility of Rabe's interpre-
tation of the origin of asteroid rotation, the probability for the process of
pair formation and evolution as he describes it is infinitesimal (Ruskol and
Safronov, 1961).
In the two- body problem the relative velocity V m of bodies before encoun-
ter, for which capture under the influence of a resisting (or nutrient) medium
is possible, is given by the energy condition
!H?f-^\FVdt, (!)
where F is the force of resistance of the medium, determined by the momen-
tum which the body imparts to the medium in one second. It is equal to the
mass of material nr z pV encountered by the body per second, multiplied by the
velocity V of the body. Consequently,
*<j
izr 2 pV*dt&nr 2 9 V*\, ^
where A is the diameter of the largest closed surface of zero velocity. For
an asteroid of radius r and density 6 = 2 g/cm 3 , Rabe obtains A — 725 r in the
Sunwards direction and A = 454 r in the perpendicular direction. He takes
the following parameters for the asteroid zone: a niri = 2a.u., a max = 3.5 a. u.,
thickness of 0.2 a. u., and total mass of matter in the zone equal to 5 ■ 10 24 g.
This yields a density of p= 10" 15 g/cm 3 . Therefore for asteroid capture it is
necessary that
V~2 ^ m
1 10 3 p » 10"
The presence of a third body, the Sun, does not substantially facilitate the
conditions of capture. Consequently the order of magnitude should be
VI < 10" 12 (2 -^ + K*,)** 10~ 12 - 2 -^. ( 4 )
Such small relative body velocities are impossible. Mutual perturba tions
between asteroidal bodies increased relative velocities to a value tty/Gm/Qr,
i. e., six orders more than necessary for capture according to (4). The
influx of bodies from the Jupiter zone (see Chapter 12) further increased
these velocities by one order. Thus the probability for pair formation
during binary collisions in a resisting medium is very low.
149
However, pairs could have formed in ternary and other encounters.
Statistical physics yields the following expression for the relative fraction
of pairs in the case of dissociative equilibrium (Gurevich and Levin, 1950)
in the semimajor axis interval da:
^ = 4(*^''"e«*VJn 1 «ta, (5)
where n t and n % are the number of single asteroids and pairs per unit volume
and a =dGm/2^, Body masses are assumed here to be uniform.
We obtain the fraction of binary systems with semiaxes a Q <Ca<&2 DV inte-
grating (5), bearing in mind that c*^°«*l:
i.«4(«« o) "'v4V (6)
For the broadest pairs Rabe takes a 2 = 360 r. Then
^- « 2 . lOW *w 10 5 P & lO" 10 . ( 7 )
n l
Owing to the very low mean density p of matter in the asteroid zone, the
fraction of asteroid pairs turns out to be very small.
Let us assume that pair formation has taken place in some way. We will
show that the probability for the pattern of pair evolution suggested by Rabe
(gradual convergence and fusion into a single body) is infinitesimal. Before
fusion can occur the pair will disintegrate in random close encounters with
other bodies (or in collisions).
Indeed, the mean disintegration time of an unstable pair (a>a ), accord-
ing to Gurevich and Levin, is given by
, _ W * v (8)
1 GnGm^a In ^1 + 4G2m2 ) 32nGpa In -^
Since
dm = 4w*Mr = «r 2 (1 + 29) pVdt t ( 9 )
we have
r = r + (l + 28) iilt. (10)
The increase in the body radius in the time t x amounts to
'.-*=™£ffa*r~«^ (11)
i. e., less than one tenth of a percent for a broad pair. However, as we saw
earlier, for a pair of bodies to combine into a single body, according to
Rabe, it is necessary that the body radii increase 3.7 times. Obviously,
this condition is practically impossible to meet. Consequently, the fraction
of asteroid pairs should be determined by the condition of dissociative
150
equilibrium, and from (7) it is negligible. For the early phases of evolution
of the asteroid belt one can assume a density p two orders of magnitude
greater than obtained by Rabe. But even so ^M^IO" 8 , i. e., the fraction of
asteroid pairs is infinitesimal.
In our opinion the rotation of the asteroids and their irregular shape can
be attributed in a natural way to direct collisions and fragmentations experi-
enced by the bodies in the course of their evolution.
151
CONCLUSIONS
The most important characteristics of the accumulation process are the
relative velocities of bodies and their size distribution. Body velocities
increase due to mutual gravitational perturbations and decrease due to
inelastic collisions. Simultaneous analysis of both factors (see Chapter 7)
reveals that relative body velocities are conveniently defined by the expres-
sion v = \jGmfir , where m and r are the mass and radius of the largest body,
respectively. If the bodies have identical masses and fuse during collisions,
then 6«sl. Given a power law of mass distribution of the bodies for the
terrestrial zone, 0^3—5. In the presence of gas 6 may amount to several
tens. As long as the dimensions of the largest bodies in a cluster did not
exceed several kilometers, relative body velocities did not exceed 1 m/sec.
Collisions among bodies took place with practically no fragmentation and
ended in fusion. This result enables us to draw the important conclusion
that when conditions permitting gravitational instability were absent in any
given zone (as was probably the case in the portion of the dust layer close
to the Sun), there could have been direct growth of bodies due to fusion in
collisions.
By studying the size distribution of protoplanetary bodies by the coagula-
tion theory method (see Chapter 8), we were able to obtain an exact solution
of the equation in the absence of fragmentation for the case where the coag-
ulation coefficient is proportional to the sum of the masses of the colliding
bodies. The mass distribution function for the bodies is a product of the
power function m~* with exponent g~ 3 / 2 by the exponential function e~ bm , which
cuts off the distribution in the large mass region. The main mass of matter
in this distribution is concentrated in the large bodies. Fragmentations of
bodies increased the amount of fine matter in the system. Qualitative study
of the coagulation equation in the presence of fragmentation makes it possible
to conclude that the mass distribution function can be approximated by a
power function with exponent q lying between jz and 2.
The power law is not sufficient to describe the distribution of large bodies.
It was obtained without allowing for features specific to their growth. Owing
to gravitation the effective collision cross- sections of the largest bodies
were proportional to the fourth powers of their radii. As a result they grew
at a relatively faster pace than other bodies and their orbits tended to become
circular. Such bodies became potential planetary "embryos." At first there
were many embryos; but as their masses and correspondingly their relative
velocities increased, the source zones of adjacent embryos aggregated. The
smaller of the embryos grew more slowly and apparently broke up before it
could land on the larger embryo.
The number of planetary embryos decreased until the distances between
them had become sufficiently large to ensure that gravitational interaction
152
would not be able to disrupt the stability of their orbits for a long time. This
determined the law of planetary distances.
As body masses grew, so did their relative velocities. Collisions between
bodies of comparable mass began to be accompanied by fragmentation. But
collisions with other bodies posed no threat to planetary embryos. Their
growth can therefore be described quantitatively in a completely satisfactory
way by assuming that all bodies colliding with them landed on them without
leading to disintegration of the embryos. Evaluation of the growth rate of
planets having separate source zones (Chapter 9) leads to a growth span of
10 years for the Earth. The Earth has long since exhausted all the primary
material in its zone, and computations of the amount of meteorite material
currently landing on it cannot be utilized to evaluate its age.
The growth process of the giant planets was complicated by a number of
important factors including fusion of planetary source zones, ejection of
bodies from the solar system by gravitational perturbations emanating from
them, and hydrogen accretion by Jupiter and Saturn (see Chapter 12).
Attempts to apply the expression for the rate of body growth to the giant
planets lead to serious difficulties. Given values of the parameter 8 as
computed for terrestrial planets and an initial surface density of solid
material 0"o computed from the present-day mass of the planets, the growth
time of Uranus and Neptune proves to exceed 10 years. This difficulty can
be resolved by taking values one order larger for 6 and Oq, which requires
one to assume that a considerable amount of solid material was ejected from
the solar system in the process of planetary growth. The ejection of bodies
led to the formation of a comet cloud at the periphery of the solar system
which is still in existence. It would seem that the total mass of ejected
bodies did not exceed one third or one half of the mass of all the giant planets
together; otherwise the latter would have been drawn appreciably closer to
the Sun due to preferential dissipation of bodies in the direction of revolution
of the planets. Such ejection corresponds to a total initial mass of the
protoplanetary gas-dust cloud of 0.05—0.06 solar masses. The loss of such
large amounts of gas from the solar system could not have taken place by
thermal dissipation. Effective accretion of gas by Jupiter and Saturn set in
after they had attained a mass of about one to two Earth masses. An appre-
ciable fraction of the gas in their zone had already dissipated by that time.
In the Jupiter zone the basic mass of volatile substances (CH4, NH3) was
in the solid state and the surface density of solid material a was several
times higher than in the asteroid zone. Condensations formed in the Jupiter
zone were 2—3 orders more massive than in the asteroid zone. The massive
embryo which developed in the Jupiter zone began to throw bodies into adja-
cent zones. These bodies "washed away" most of the bodies in the asteroid
zone, increasing the relative velocities of the bodies that remained. When
the energy of relative motion of the bodies in that zone had become substan-
tially larger than the potential energy at their surface, collisions among
bodies began to result in fragmentation rather than fusion. Thus in the
asteroid zone the accumulation process came to a standstill at an inter-
mediate stage. The growth of Mars was also slowed down by bodies ejected
from Jupiter's zone. Initial condensation masses were larger in the Saturn
than in the Jupiter zone. The Jovian embryo overtook Saturn's embryo, when
the latter had become comparatively large and the influx of bodies did not
present a threat. In the Uranus zone condensation masses were large and the
surface density was lower, the uranian embryo developing more slowly.
153
Incoming bodies from the zone of Jupiter (and later Saturn) were unable to
interrupt the accumulation process, but they succeeded in slowing down
Uranus' growth somewhat by comparison with Neptune, and were responsible
for the anomalous inclination of its axis of rotation.
The bodies that landed on the planets in the process of their growth
imparted rotational momentum to them. The general motion of the entire
system of bodies around the Sun caused the bodies to impart to the planets
a regular angular momentum component — direct rotation. In addition each
individual body, having a random direction of relative velocity, also imparted
a certain random angular momentum component. For the same mass of
incident material, the smaller the bodies, the greater their number N; con-
sequently, the smaller the bodies, the better the averaging of the imparted
angular momenta and the smaller the mean value of the random component
(which is inversely proportional to the root of N). Since the random angular
momentum component was responsible for the inclination of the planets' axes
of rotation, the dimensions of the largest bodies landing on a planet can be
evaluated from the observed axial inclination (see Chapter 11). Computations
show that the masses of the largest bodies that landed on the Earth amounted
to about one thousandth of an Earth mass. The anomalous rotation of Uranus
is due to the fact that its random component of rotation was greater than its
systematic component owing to the large size of the bodies landing on it. The
masses of the largest bodies that landed on Uranus amounted to nearly one
Earth mass.
Analyzed by themselves, the equations of angular momentum and energy
conservation on transition from a cluster of bodies and particles to a planet
cannot furnish an explanation for the direct rotation of the planets. Direct
rotation is not a result of large thermal losses in the process of planetary
formation, as assumed by Shmidt. Rotation is determined by concrete condi-
tions of collision among merging bodies, i. e,, by the fundamental laws
governing their motion, and it can be determined by statistical analysis of a
limited three-body problem. Attempts to solve the problem numerically are
encouraging.
Thus the theory of planetary accumulation from solid material furnishes
us with a natural explanation, based on a unified point of view, of the princi-
pal laws of the solar system and of such characteristic features as the
presence of the asteroid belt and the anomalous inclination of Uranus' axis.
However, the absence of a growth theory for the giant planets which would
account for all principal features of the accumulation process in their zone
prevents us for the time being from giving definite answers to a number of
questions, such as: How much substance was ejected from the solar system?
How long did the outer planets actually take to develop? What were relative
body velocities in this zone? How large were the largest bodies in the asteroid
zone? Further progress in accumulation theory (especially the construction
of a quantitative growth theory for the giant planets) and more extensive
utilization of various kinds of observational data to check theoretical results
are among the tasks now facing planetary cosmogony.
5979 154
Part III
PRIMARY TEMPERATURE OF THE EARTH
Chapter 14
INTERNAL HEAT SOURCES OF THE GROWING EARTH
AND IMPACTS OF SMALL BODIES AND PARTICLES
36. Warming of the Earth due to generation of heat by
radioactivity and compression
By the primary temperature of the Earth we mean its temperature at the
end of the formation process, which lasted over a period t^IO years
(see Section 27). To evaluate this quantity it is necessary to consider the
three main sources of heat of the growing Earth: 1) impacts of falling bodies;
2) generation of heat by radioactivity; 3) contraction of matter due to the
pressure of the layers being added at the top. The warming due to
compression of matter was investigated by Lyubimova (1955). It was found
that the temperature rise in compression is proportional to the temperature T
of the compressed matter and is given in terms of the Griineisen coefficient 7:
*T = iT±, (1)
where 7 = ^-*, a is the volumetric coefficient of thermal expansion, and A", is
the adiabatic bulk modulus.
For the Earth's mantle y was approximated by the expression y^ 2 3. 5 p~ 2 .
It was assumed that the core consists of metallized silicates. The distribu-
tion of density inside the Earth just before the phase transition (for M = 0.8 Q)
was approximated by the Roche formula. Assuming that compression of the
phase transition itself (discontinuity in the core) was not accompanied by
warming, Lyubimova obtained the ratio f=T/T for different depths. Compres-
sion of matter causes the initial temperature to increase by roughly 2.3
times at the Earth's center and by 1.8 times at the boundary of the core.
Warming of the Earth by radioactive heat in the course of its growth was
disregarded.
In the presence of sources of radioactive heating, an additional term edt
must be inserted in expression (1):
dT=jT^- + edt, (2)
where the quantity e (radioactive warming per unit time) can be considered
constant. The equation must satisfy the initial condition
T(r t t(r))=T $ (r) for «=t(r), (3)
155
which means that at the instant t (r) when the radius of the growing Earth
equals r, the temperature of its surface will be T t (r). The contribution due
t*o radioactive heat is maximum in the central region of the Earth, which
was the first to form. The Earth's compression is also maximum in the
center. But the initial temperature T, is minimum in the central part. A
calculation of T 9 =T, for an Earth bombarded by small bodies and particles,
as well as an approximate calculation of T with the aid of / from the formula
Mr)«[r., + .(T,-*(r))]i + /
(4)
for e= 300° in 10 years was carried out by the author in 1958. The curve
for T 1 (m 1 ^) is given in Figure 9.
r,
woo
800
too
(fOO
200
at au as o.Bfm/Q) f / 3
FIGURE 9. Primary temperature of Earth
resulting from impacts of small bodies
and particles in the accumulation process
(dotted lines) and from warming due to
compression of Earth material and gene-
ration of radioactive heat. Solid line
shows the Earth' s temperature 100 mil-
lion years after it began to grow, allow-
ing for all three heat sources for £=300°
over 10 8 years.
T^
N^y
-
1
^A
tt \
The Griineisen coefficient has recently been evaluated from more recent
data by Lyubimova (1968).
For the mantle
T = Ti = -t? — *k. «, = 6.72, 6 1 = 0.13,
for the metallized silicate core
(5)
T = T» = &» — **P. H = 0.146, b % = 2.18.
Correspondingly equation (2) for the mantle becomes
(6)
(7)
156
For the initial condition (3) its solution is
T{t " »HrfcT$' e °' { *~^ T '+ e '^\ te ^( ! ^T dt '- (8)
h
Here T (/ t , t) designates the temperature at the instant t of a spherical layer
formed at the instant t t . Similarly T (m t , m) will denote the temperature at
the surface of a sphere containing the mass m a for a planetary mass equal to m.
The nature of the Earth's core has yet to be established (Magnitskii, 196 5).
In experiments on compression by impact the corresponding transition is not
detected before pressures exceeding 4 million atmospheres are reached.
But from a cosmogonic point of view an iron core would pose greater difficul-
ties than a metallized silicate one (Levin, 1962). Below we will consider a
terrestrial model with a silicate core. If the core were of iron, it would
have to be formed distinctly later than the Earth itself. The initial tempera-
ture would then have to be evaluated for a coreless Earth, with 7^^ for the
entire Earth.
The warming of the core material is given by expression (2) with the
Gruneisen coefficient Yi before the phase transition and by the same expression
with the coefficient y 2 afterward. According to Ramsey (1949), contraction
during the phase transition takes place with practically no warming (the
energy is transformed into the work of deformation). This problem also
awaits definitive solution, and Ramsey's assumption is best seen as a
variant yielding a minimum value of the temperature. Solving equation (2)
separetely for different y and assuming that the phase transition by itself is
not accompanied by warming, we obtain the following expression for the
temperature inside the core:
-f e—*VM. t) J «*,,<*„ n CJ'-^Y 1 edt>. (9)
Here p_ and p + are the density just before and after the phase transition and t*
is the time corresponding to the phase transition at the point under conside-
ration. For all m 8 ^ 0.08 Q the phase transition occurs when m & 0.8 Q,
while for the remaining values of m t inside the core it occurs later.
Alternatively, one could assume that the phase transition generates the
same amount of heat as given by relation (2), with 7 assuming intermediate
values between y 1 and y 2 . If as in (6) we take a linear dependence of 7 on p
during the transition, we obtain
T = 1.58 — 0.086p. (10)
An expression similar to (9) is then obtained for the temperature inside the
core, except that the first two terms on the right are 1.8 times larger.
Indeed, if we assume that the effective y responsible for warming in phase
157
transition amounts to only a fraction C<1 of the value (10), then the correction
factor in the first two terms of (9) will accordingly be 1.8 C .
In expressions (8) and (9) it is convenient to take the mass m of the
growing planet as independent variable instead of the time t. These quantities
are related by (9,14):
*L.*«i^ja v .(, —2-)=^ (i+ap.."'^-?-), (id
where p=p (m) is the mean density of a planet of mass m. Relation (8) for the
temperature inside the mantle can then be written as
T («., w)««*(fc~rt^))(— A— Y 1 T +
^ J Lf(«.. m)J {dmldt)'-
= A x {m„ m)T, + B x (m, t m)t(t-t % ). (12)
Expression (9) for the temperature of the core is modified in a similar way.
Inserting the factor 1.8 C for possible generation of heat in the phase transition,
we obtain the following expression for the core:
•Nfl
where m* is the body mass for which phase transition occurs at the point m t
(i. e., at the surface of a sphere containing a mass m t ).
Quantity T (m # , m) has been calculated by the author from formulas (12)
and (13) for values of the density p (m §f m) at different points m t inside a body
of mass m and mean density p(m), taken from data given by Kozlovskaya
(1967). Kozlovskaya computed a series of body models of different mass on
a BESM-2 high-speed computer, for an equation of state corres-
ponding to undifferentiated terrestrial material. It was assumed that po — 3.47,
P.=» 5.41 and p + = 10.16, which corresponds closely to model No. 7 for the
Earth according to Pan'kov and Zharkov (1967). In this series of models the
mean density of the bodies depends on their mass as follows:
0.1
0.3
0.5
0.7
0.838
0.838
0.9
1.0
3.47
3.70
3.99
4.16
4.31
4.39
4,71
5.08
5.54
Numerical integration of (11) with these values of }(m) yields the time
dependence of the growing Earth's mass depicted in Figure 10. The time
required for the Earth to grow to 97% of its present mass for 6— 3 is
5979
158
86 million years, which confirms the value 88 million years estimated for
the same assuming a constant density intermediate between the initial and
final density (p« 4.5). For 6=5 the duration of the growth process decreases
to 55 million years.
(m/Q) f ? 3
/.o
8*5 /
/f'J
05
r I
lilt
20 (tO SO 80 fOO
t , million years
FIGURE 10. Rates of growth of the Earth
for two values of the parameter charac-
terizing relative velocities of bodies.
H.O
K-to
3.0
-
0.5
A
^ s
2.0
^^T"
3
^ K~to
0.5^
JL.>l_ ^^^
to
0.5
10 i
FIGURE 11. Dependence on m,of coefficients A (solid
line) and B (dash-dot line) characterizing terrestrial
warming due to contraction and to the decay of radio-
active elements.
Figure 11 illustrates the dependence on m t of the coefficients A and B t
defined according to (12) and (13) (subscripts have been dropped for brevity)
and computed for m =0.97. Table 15 lists the values of the second term
Be (t-t 9 ) for e= 200°, t - 10 8 years, and 6- 3 and 5.
TABLE 15
m.
c
1
1
1
0.15
0.29
0.35
0.50
0.65
0.8O
0.90
0.97
6^3
224
158
139
133
115
96
70
48
0.1
234
165
147
0.5
282
195
185
1.0
357
245
248
8 = 5
143
101
89
85
73
61
45
27
0.1
149
105
94
0.5
179
125
118
1.0
227
156
158
It is evident from Table 1 5 that radioactive heating played a relatively
unimportant role during the period of terrestrial formation: the inner portion
159
of the mantle was warmed by roughly 100°, the core by 150 — 200°. Of
greater significance is the warming due to the heating T 9 of the surface of the
developing Earth by impacts from falling bodies and to the A -fold rise in
this temperature resulting from compression of material in the Earth's
interior. For m^O.5 one can assume that the minimum value of T t is given
by r f «350°. Then AT § & 600°, i. e., it exceeds the heating Be (t-t,) due to
radioactivity by a factor of five or more. Therefore if one wishes to arrive
at a more accurate value for the initial temperature it is most important to
correct the value of T t , the temperature at the surface of the developing
Earth.
The energy spent in warming the Earth due to contraction of its material
is small compared with the total energy of compression. Energy is expended
chiefly in the deformation of material. Lyubimova (1962) has evaluated the
energy expended in the elastic deformation of a homogeneous sphere of
terrestrial mass under the influence of its gravitational field. According to
the model used in the calculations, a nongravitating, undeformed Earth is
formed initially; its gravitational field is then included, with the corres-
ponding deformation. Depending on values used for the parameters, the
estimated energy of deformation varies between the limits (5 — 9) • 10 38 erg,
i. e., it amounts to a considerable fraction of the potential energy of the
Earth as a sphere (Lyubimova, 1968). If an appreciable fraction of this
energy had been contained in shearing stresses and had been liberated in the
process of the Earth's evolution upon relaxation of these stresses, it could
have been an important source of internal energy. In reality the Earth's
development was gradual as was the intensification of its gravitational field,
and the deformations increased gradually. Allowance for this should yield
a smaller energy of deformation. Another element of inaccuracy in the
calculations is the insufficient reliability of the numerical values used for
the parameters characterizing the elastic properties of the Earth's material,
making it impossible to draw definite conclusions.
In this connection it is of interest to calculate directly the energy of
compression of the Earth during its growth, which represents that fraction
of the Earth's energy of deformation which cannot be liberated in the
relaxation of elastic stresses and cannot convert into heat. To evaluate the
energy of compression it is sufficient to know the equation of state P (p) and
the density distribution inside the Earth. Quantity P (p) can be approximated
in the form
P = af — b. (14)
Then the energy of compression per unit mass is given by
and the total energy of compression by
W = 4* \W( ? ) 9 r*dr. (16)
The approximation of the curve of P (p) obtained by Kozlovskaya (1966) for
p = 3.47 gives the following values for the mantle: n = 18 / 8l o= 1.05 • 10 9 , and
b= 2.4 * 10 11 . Integrating (16), we then find that the energy of compression
160
of the mantle material is W m air= 0.6 • 10 38 erg. The material of the silicate
core obeys the same equation of state as the mantle material up to the phase
transition, after which one can take an equation of the form (14) with re'= 3,
a'= 2.2 • 10 9 , and b' = 1.0 • 10 12 . The total compression energy of the core
material is found to be 3.6 • 10 38 erg. Of this, 0.9 * 10 38 erg goes for
compression before phase transition, 2.3 • 10 38 erg for compression during
and 0.4 * 10 38 for compression after the phase transition. Thus the entire
compression energy of the Earth's material amounts to
^^Wman+^core^O^.lO^ + S.G.lO^^^.lO 38 erg. (17)
The total energy of deformation should not be less than the above energy
of compression. It is interesting to note that over half of the compression
energy is expended in phase transition in the core and only x / 7 in contraction
of the mantle material. Thus we see how important it is to know the change
in thermal energy which occurs in the phase transition. The conversion
into heat of a mere 10% of the energy expended in the transition would have
led to warming of the core material by 1000°.
The foregoing estimates were based on the assumption of a silicate core
that has gone over into a metallized state. The problem of terrestrial
thermal processes must be stated differently if one assumes an iron core.
According to an estimate by Lyustikh (1948), about 1.5 * 10 38 erg should
convert into heat when iron overflows into the core from the mantle,
corresponding to warming of the material of the whole Earth by 2400°.*
Assuming that this overflow was possible (which would require the existence
of large inclusions of metallic iron), it must have occurred after the Earth
had formed, when it had warmed up sufficiently. This applies to the differen-
tiation of all substances in the Earth, the total energy of which may have
been considerable (see, for instance, Krat (I960)) and should be taken into
account when studying the thermal history of the Earth. Unfortunately there
are no definite data on the scale of differentiation.
37. Warming of the Earth by impacts of small bodies
and particles
Formulas (13.12) and (13.13) for the initial temperature of the Earth
contain the temperature T a of the surface of the growing planet. It is
determined by the energy of impacts from bodies falling on the Earth during
its formation, and moreover depends on the dimensions of these bodies. The
simplest way of estimating T 9 is by assuming that the Earth was formed
from small bodies and- particles. We will denote the surface temperature
for this case by TV Bodies can be regarded as small in the problem under
consideration if the energy liberated when they fall on the Earth is liberated
* Owing to the low rate of settling of the heavier inclusions, their kinetic energy is negligible (even for a
viscosity of 10 17 poise and inclusion radius of 100 km, the Stokes velocity will not exceed 1 cm/sec). The
potential energy liberated by the inclusions as they settle down should therefore convert into heat throughout
the Earth* s sphere, without leading to preferential warming of the core compared with the mantle. An iron
core impoverished in radioactive elements could not subsequently become warmer than the lower mantle.
This makes it difficult to explain the Earth' s magnetic field., which is usually related to convective motions
in the liquid outer part of the core.
161
in the immediate vicinity of the surface and is almost entirely irradiated
into space. A layer of thickness h warmed on impact will cool within an
interval of the order of h % ik, where k is the coefficient of thermal conductivity.
The time required for the laying down (due to the Earth's growth) of a new
layer of material of thickness h is given by hit. If the latter is less than the
former, the greater part of the layer's heat will remain inside the Earth.
Bodies can therefore be termed small if the thickness of layer warmed upon
their settling is
A<*/r. (18)
From (9.14) it can be shown that the rate of increase in the Earth's radius
/*~10~ cm/sec. For the usual molecular thermal conductivity Jc^-ICT 2 , we
obtain h ^ 1 km. Since the thickness of the layer warmed on impact is of the
order of the diameter of the fallen body, it follows from (18) that all bodies
with diameters less than a hundred meters can be classed among small
bodies; the energy of their fall is almost entirely emitted into space. In
Section 40 it will be shown that due to the considerable mixing of the material
by impacts from falling bodies, the effective thermal conductivity was 2 — 3
orders greater than the molecular thermal conductivity. The dimensions of
the bodies whose energy of fall was trapped inside the Earth must have been
correspondingly larger as well.
Suppose the Earth's growth took place as a result of the fall of small
bodies (in the above sense) and that the energy they imparted was liberated
practically at the surface. Without introducing a serious distortion we can
assume that the rate of increase of the Earth's radius t was constant and
that its surface was flat. In this case the surface temperature T t0 is
independent of the time and can be determined from a simple relation
expressing the equality of the energy brought by the bodies and the energy
emitted:
(T i +T■)^ = WV ( r J.- 7, J) + c ( 7 '-)-^)^■• (19)
Here m and r are the mass and radius of the growing Earth, T p the tempera-
ture of the falling bodies and particles, v their mean velocity with reference
to the Earth before encounter, T the black- body temperature near the Earth,
a' the Stefan- Boltzmann constant, and c the heat capacity. Temperatures are
reckoned from absolute zero. The left-hand side of (19) represents the
energy imparted to the Earth by falling material per unit time. The first
two terms on the right represent the energy lost by the Earth due to emission
(emission minus absorption). The last two terms on the right are the energy
expended in warming terrestrial material. They are nearly two orders
smaller than the others and can be totally disregarded, since the main term
in (19) contains T^ in the fourth power. Substituting for dmidt from (9.14)
and inserting v*=Gm/Qr, we obtain (Safronov, 1959)
i «o— 'o"! 26a' Pr ' ^ U '
162
The values of 7*^ obtained from this expression for different m are given
in Figure 9. They are maximum for layers now situated at a depth of 2 — 2.5
thousand km, and for 0=3—5 they amount to about 350 — 400°K.
The low gradient of T, 9 over r validates the presumption of stationariness
and means that the surface temperature Ta of the growing Earth was simul-
taneously the temperature of its material before appreciable amounts of heat
had been liberated inside it due to radioactivity and contraction.
163
Chapter 15
WARMING OF THE EARTH BY IMPACTS OF LARGE BODIES
38. Thermal balance of the upper layers of the growing Earth
When bodies settle on the Earth, most of the energy of impact is liberated
inside a layer having a thickness of the order of the diameter of the fallen
body. The surface temperature fluctuates sharply in the process, and its
mean value is slightly less than the value obtained earlier for T t0 , since
jPsur<> / T|^ r = T& - In order for heat to escape from the layer warmed by the
impacts into the open, there must be a negative temperature gradient along r.
Consequently, the thicker the layer, i.e., the larger the falling bodies, the
higher the temperature of the material under the layer. On the other hand,
the larger the fallen body, the larger the crater it produces and the greater
the depth of mixing during impact. Heat transfer by mixing of material
during the fall of large bodies is far more efficient than heat transfer by
ordinary thermal conduction (molecular, radiant, etc.). As there is no
theory yet which would permit us to allow for the specific character of
mixing by impact, it is natural to seek to use the methods of the theory of
heat conduction. To do this we must determine the appropriate value of the
analog of the coefficient of thermal conductivity K associated with mixing,
and the depth distribution of the heat sources £ .
On the whole the problem of the Earth's warming by impacts of falling
bodies is fairly complicated, as it involves setting up and solving the
equation of thermal conduction (more precisely, of heat transfer) for a
spherical volume having a moving boundary with an adjacent region of
heating and intensive mixing. To determine the quantities K and £ entering
into this equation, one must in turn know; a) the shape and size of the
craters formed during impacts; b) the fraction of energy expended in warming
the material under the crater and its depth distribution; c) the mass distri-
bution of bodies from which the Earth was formed.
The data available on these questions are unfortunately highly unreliable.
In particular, at present there exists no sufficiently complete theory of
cratering, and almost nothing is known of the results of the fall of very
large bodies, where the force of the Earth's gravity disrupts geometric
similitude substantially, leading to qualitatively new phenomena. We will
therefore have to confine ourselves to the simplest schemes if we wish,
first, to evaluate the role of the main factors in the first approximation and,
second, to determine whether the falling of large bodies on the Earth could
have led to an appreciably higher initial temperature than obtained in
Chapter 14.
164
Owing to the random character of impacts due to the infrequently falling
large bodies, K and S varied sharply in space and time, leading to uneven
warming of the Earth. The question of initial thermal inhomogeneities is
discussed in Chapter 16. In the present chapter we consider only the mean
values of K and &, calculating therefore a mean smoothed initial temperature
of the Earth. Our first step will be to derive an initial equation and obtain
its solution; next we will seek to calculate K and £.
Below it will be shown that K varies appreciably with depth. Although the
general equation of thermal conduction for K dependent on x remains linear
as before, in practice it is complicated to obtain a solution satisfying the
required initial and boundary conditions, especially if the boundary is moving.
In calculating the warming of the upper layers of the growing Earth by
impacts from falling bodies, it is therefore expedient to confine oneself to
the simpler case of the stationary state of a plane half- space whose boundary
is moving with a constant velocity dr / dt —t.
In a reference system bound to the material, the equation of thermal
conduction has the form
where x is reckoned from some initial position of the surface. Quantities K
and & are functions of the distance z from the actual position of the surface:
K = K(z^ £ = £(«), z=x+rt. (2)
It is therefore natural to pass from the independent variable x in equation (1)
to the independent variable z. Then
T(xt) = r(ztY d -l- d Jl + f^- ^=^ (3)
Obviously, the steady solution should also be sought in a moving coordinate
system: dT*\dt= 0. For brevity we drop the asterisk (T*(z, t) = T(z)). Then the
transformed steady equation of thermal conduction takes the form
*g+(£-')£+*=°- (4)
The solution of this equation should satisfy the boundary conditions:
7'(0) = 7 , f0 for z =
dT(oo)/dz = for z = cx) (5)
Let us set
*£ = «• (6)
Then
u' — -^u + g = (7)
165
and
where
00
*=j£*.
(8)
(9)
Owing to the second boundary condition, u(oo)« and therefore C— 0.
Let us now calculate 7*:
00
T = j -£- dz = j JTVUs j e-^dz + C.
r
From the first boundary condition,
T = J K-'e*dz j e-rgdt + T M .
(10)
Expressing z in terms of y and changing the order of integration, we reduce
the double integral to single integrals:
* 00 f 00
\ K-Vdz \ er*>$ (*') dz' = i [ My J *"*'<? tf) K (y>) dy> =
J r*SW)K(tf)dtf \*dy+\ e-*'S(y , )K(y>)dy>\e*dy
-° Of o .
"? °°
Lo * J
Passing from y back to «, we obtain the expression for T;
7 (*):=}! J (l-e-')Sdz + (*-l)j^«*l + f , .r (U)
The primary Earth temperature T a due to its warming by the impacts of
falling bodies is clearly
00
T, = T (oo) = 1 J (1 - f) gdi + T<
(12)
39. Fundamental parameters of Impact craters
In order to calculate quantities K and S in equation (12), it is necessary
to know the size of the crater formed by the fallen body, the thickness of
the layer settling around the crater, and the depth distribution of the energy
liberated on impact. We will assume that the fall of a body of radius r' will
166
form a cylindrical crater whose "original" depth h and "original" radius R
are proportional to r':
A = v/ f /? = v 2 r'. (13)
The term "original" refers to the size of the crater before its sides collapse
and before part of the ejected material falls back into the crater. Below we
will show that for a uniform rate of fall, v x can be considered constant while
v a decreases slowly with increasing r'. Salisbury and Smalley (1964) assume
on the basis of laboratory studies (Gault et al., 1964) that for an impact
velocity of 11 km/sec the ejected mass will be 10 3 times greater than the
mass to' of the fallen body. Crater depth at this speed amounts to about two
diameters of the falling body (Opik, 1958; Bjork, 1961; Andriankin and
Stepanov, 1963). Consequently one can take v^4 andv 10 ^18, where v M is the
value of v a for small r'.
Suppose further that the material thrown out of the crater evenly covers
an area in a circle of radius R x with a layer of thickness h x :
K=f t h; (14)
R x is determined by the rate of ejection of material from the crater.
Assuming that in the propagation of shock waves the energy of motion is
constant and identical in all directions (Afv*= const, where M is the mass
affected by the explosion), Stanyukovich (1960) obtains the following distri-
bution of the velocity of the ejected matter:
'-*£)*. <»>
where v is the impact velocity of a meteorite of radius r'. The velocity v
characterizes all particles on a ray of length R'=\jR*-\-w* beginning at the
center of the explosion at depth w and ending at the surface at distance R
from the epicenter. Arguments in favor of this result for large impact
velocities are also cited by Andriankin and Stepanov (1963). On the other
hand, Lavrent'ev (1959) and Pokrovskii (1964) hold that the momentum
remains constant and vccR r *. Basing himself on his solution of the problem
of concentrated impact for a single simplified model, Raizer (1964) concludes
that the velocity law is appreciably closer to RT* U in this case than to if' -3 .
Dokuchaev, Rodionov and Romashov (1963) give the velocity law vocit' -1 - 8 ,
obtained by measuring the maximum rate of dispersion of particles as a
cupola rises over the site of a deep explosion. This law is also closer to the
case of energy conservation than to that of momentum conservation. For the
impact velocity of interest to us (10— 12 km/sec) the amount of material
which evaporates is comparatively small and the energy of expansion of the
resulting gases plays a relatively unimportant role. Thus Shoemaker (1962)
found from data given by Altschuler regarding the equation of state of iron
that the main meteorite mass melts only when the impact velocity reaches
9.4 km/sec. But as far as intensity of pressure is concerned, impacts
having such velocities are similar to explosions intermediate in character
between chemical and nuclear explosions.
167
At low explosion depths w the dispersion rate v obtained from the relations
of Dokuchaev and others (especially Pokrovskii) is too high. In this respect
expression (15) is to be preferred. Let us take
»=»•(£)'•
(15'
The total energy of disperion of material for a conical crater of depth hz&w
is given by
^=ST^ = *i\^TP-^=f4^^- 3 xf
»*** :7]m ^. (16)
u w
From this we obtain the ejection efficiency coefficient tj:
where n—Rlw is the ejection index. Table 16 lists numerical values oft).
TABLE 16
Q
n
1.5
1.8
3.0
2.0
4.5
u) = 4r'
0.28
0.39
0.10
0.12
0.002
0.002
w =
3r'
2.0
0.28
0.12
0.004
4.5
0.39
0.15
0.005
The rows with n= 4.5 and n= 2 apply to the fall of small and large bodies,
respectively. According to Dokuchaev and others, in deep explosions in
clay n increases from 0.11 to 0.14 as r\ varies from 1.7 to 2.8 (see Figure 47
in their work). Closest to these figures in Table 16 are the values of r\ for
q= 1.8.
Expression (15') makes it possible to evaluate the distance R t of maximum
dispersion if the direction of the dispersion velocity is known. Usually the
initial direction of the velocity from the center of the explosion is taken to
be radial (as an approximation). According to Dokuchaev and others, for
deep explosions (n^l.5 — 2.5) the initial velocities of points on the surface
are directed, in the first approximation, radially from a point situated at
twice the explosion depth (2w). However, the authors believe that their
conclusions cannot be extended to explosions with relatively little hollowing
(i. e., with large n). We will take as i?! (approximately) the distance of
dispersion of particles whose radii vectors are directed from the center of
the explosion at an angle of 45° to the horizon. Then
*— + Aji -+7-+^r-^+ f -di
vf) 8 *'
(18)
168
For a vertical impact with a velocity of 10— 12 km/sec one can take
v!«j4; for an impact at an angle of 45°, Vi&Z. Then for q= 1.8 the maximum
distance AJ? from the take-off point is found to be 24 and 68 km, respectively.
The dimensions of the crater are determined by the energy and depth of
the explosion. With regard to the depth of penetration of the body w
("explosion center"), and therefore the depth h of the original crater, which
is slightly larger than w, geometric similitude holds: for the same impact
velocity they are proportional to the radius r'of the falling body (i. e., to
the cube root of the impact energy). Therefore the "relative explosion depth'
wjC 11 ', as it is usually taken in blasting (with the power 1 / 3 on the charge C),
is independent of the size of the falling bodies and for w=4r' and velocity
v = 11 km/sec it equals 0.07 m/kg v », which corresponds to small, near-
contact explosions. The power 1/3, however, is probably suited only to very
small craters not more than a few meters deep, where the energy which
must be expended in the formation of the crater is proportional to its
volume R*w=n*u? (overcoming atmospheric pressure, destruction of matter).
In the case of large bodies, an additional, considerable amount of energy is
expended in overcoming gravity to lift the material out of the crater. This
energy is proportional to R z W'W~n 2 u^, and its relative importance increases
with increasing wccr'. The geometric similitude with respect to the crater
dimensions is therefore destroyed. On the basis of extensive data on
chemical and nuclear reactions, American specialists have concluded that
the linear dimensions of the crater increase in proportion to the explosion
energy in the power 1/3.4 (Shoemaker et al., 1961; Nordyke, 1962).
Assuming radial dispersion and disregarding the resistence of the material,
Pokrovskii (1964) found, from the condition that the ejection rate along the
crater slope was such that the material was thrown out only over the
crater's edge, that such craters (with the same n = Rjw) are obtained for
constant fi>/C l/3 - fi . From experiments concerning the expansion of a gas
bubble in sand contained in a vacuum, Sadovskii, Adushkin and Rodionov
(1966) concluded that n must depend on wjC % . For an exponent 1/3.4,
noc^) 3 - 4 =(0 « . Pokrovskii's relation gives n oc (r 1 ) 8 for n 2 >l. The
relation of Sadovskii and others is logarithmic. It was obtained for the
interval 0.6<n<2.6, and its form for large n of interest to us is unknown.
One can apparently take race r*"* for r'^^ and
p- = 'i = *o(2)' ^r H>r , (19)
v 1 = v 10 for r l <> ,
where a-< 1 and r is of the order of a few meters.
Since the relative explosion depth should be determined by the ratio w t C VVm ,
where 3 <^< 4, it should increase with the size of the falling bodies: w oc r';
I i--
Ca r' 3 and wjC * or (/-') * . As r' increases, the explosions become relatively
deeper and n decreases. For large enough r'^r^a. "loosening explosion," in
which practically all the material thrown out falls back into the crater, may
take place. If the function n = f(-^\ extends to explosions that qualify as
relatively small (with respect to the magnitude of w\Ci*), it is easy to
determine how much one needs to add to the absolute depth w in order to
obtain explosions with a specific n for a small value of wjC 11 *:
169
■-/(£).-*(£)-'.(*■*"')• (>•>
At depths 01 10— -30 m loosening explosions occur when wjC^l. For
falling bodies this ratio is 15 — 20 times smaller. Therefore to obtain the
same value of n, w^' 1 should, from (20), be increased as many times. For
fi= 4 the value of n will be the same for falling bodies as for loosening
explosions with u>/C v, = 1 and w=20m provided w& 60 and 150 km, and if
v x = 4 and 3. Impacts similar to loosening explosions should therefore take
place on the falling of bodies with radius ^^15 and 50 km, respectively.
In instances where n= 3.4 the size of bodies which produce impacts similar
to loosening explosions should be many times larger. However, one can
expect that at such considerable depths the role of gravitation becomes
dominant, with \i approaching 4. Baldwin (1963) concludes on the basis of
studies of the parameters of lunar craters that \i increases with the size of
falling bodies, reaching 3.6 for craters 10 miles in diameter. But his
claim that n begins to decrease for yet larger craters is unfounded.
The condition for transition to loosening can be obtained directly from
energy considerations. The ejection from a crater of the material of mass AT
contained inside it involves expenditure of the energy
E e ^Mgh,^Mg%^fMgr', (21)
where h p &w/2 is the minimum height to which the center of gravity of the
mass M must be raised for it to be thrown out of the crater. Since
£ ej =7]m'i>;/2 and Af/m'«2Sft*£i
(for a spheroidal crater), the size of a body forming a crater with index n
should satisfy the condition
where r is the Earth's radius. For n^^ an ejection explosion will grade
into a loosening explosion. Therefore
'!<£■
(22)
It is usual to take n x ml, and it seems one can assume that n^O.7. Then for
the present terrestrial radius and v l ^ 4, one obtains /-^(lOO — 200) r\ km,
while for v^ 3 it is three times greater. If departures from geometric
similitude due to the important part played by gravity cause the impacts of
large bodies to resemble deep explosions in all respects (in contrast with
small body impacts, which are similar to contact explosions), the ejection
efficiency coefficient can be taken to be about 0.10 — 0.15 (according to
Dokuchaev). Then r^lO— 30 km for Vl = 4 and ^30 — 90 km for v x = 3.
These values agree entirely with results obtained earlier from other
considerations.
170
Thus one may conclude that for falling bodies with radius r'>r v where r x
equals a few tens of kilometers, the impacts will be similar to loosening
explosions. Impacts with loosening are of interest inasmuch as they warm
the Earth most effectively (nearly all the heat of the fallen body remains
buried inside the filled- in crater).
40. Heat transfer in mixing by impact and depth distribution
of the impact energy
The fundamental relation defining the coefficient of thermal conductivity AT
is the well-known expression which relates it to the flux of heat:
H(x) = K{x)%. (23)
Here for the sake of convenience H (x) denotes the "temperature flux," which
differs from the heat flux by a factor 1/cp and is directed upward toward the
surface (i.e., toward decreasing x). Thus the evaluation of K can be reduced
to the calculation of the heat transported across a unit surface at depth x
during impacts from falling bodies.
Consider first the effect of a single impact leading to the formation of a
crater of depth h. In (23) x is reckoned in a fixed coordinate system. Since
K and T depend on the distance z to the surface, which shifts with time due
to the Earth's growth, it is convenient to choose x to be identical with z at
the instant of impact. Henceforth we will write z in place of x but evaluate
the flux across a surface fixed in the x system. This precaution is
unimportant in practice, as the mean rate of displacement of the Earth's
surface (from which z is reckoned) is 2 — 3 orders less than the rate of
filling of the crater (the ejected mass being 2 — 3 orders greater than the
mass of the incident bodies).
In the formation of a crater of depth h, the amount of heat (difided by cp)
transported upward together with the ejected material across an area of
1 cm 2 at depth z is given (assuming a linear march of T (z)) by
(fc _ I)r (*+i)«(k-„[r W +»^g]. (24)
The amount of heat carried inward across the same area during the filling-up
of the crater (without the energy £ imparted by the impact) is given by
(h — z)T (V/2), (25)
where T (V/2) is the mean temperature of the material filling the crater,
i. e., averaged over all other craters (different depths h') contributing
material to the filling of the given crater. The resulting heat transported
across a small area at depth t due to cratering at a depth h>z is given by
Cfc-«)[r(*)-f(*72) + ^g]. (26)
171
To evaluate T(h'l2) it is necessary to know the rate of formation of craters of
various sizes and the area covered by ejected material.
Let n(r') be the distribution function of bodies falling on the Earth, i. e.,
the number of bodies per unit interval of radius r 1 falling over the entire
Earth per unit time. For the power distribution law n(r') = C f r'~ p the fraction
of Earth surface covered per second by craters produced by bodies with
radii between r' and r'-\~dr' is given by
^ = ^^(0^=^.'^-^', (27)
where r is the Earth's radius. Here s represents the mean frequency of
ejections of the given scale at any point on the Earth's surface. The constant
C f can be expressed in terms of the rate of growth r of the Earth's radius:
^ = 5„(r')A« S W = 4^rir' = W^
and
C , = 3 J1 _^ > (28)
where 8 and V are the mean density of the Earth and of the bodies incident
on it, respectively.
The ejection due to a body r' will blanket an area itRz — nR 2 outside the
crater R with a layer of thickness h } (see (14)). The rate of blanketing by
such ejections outside the craters produced by them is given by
*A' = %^V')^'. (2 9)
The mean value T(h'f2) for a filled- in layer of thickness k — z can be
written as
f (A7 2,«-!L_ , {30)
I
h-itidr*
where the upper limit \(h — z) is determined from the condition that within the
time of filling of the layer k — z, no body larger than \(h — z) actually manages
to fall. One could take the mathematical expectation for such a fall to be,
for instance, 0.5. Then \ can be determined from the relations
\ h lSl dr f ti=h-z; j Sl dr'At = 05. ( 31 )
In "loosening" impacts, which occur for r f >r l , the calculation of K(z) is
slightly different from its calculation for ordinary cratering impacts. The
transition from cratering to loosening is gradual. But to simplify the
172
calculations we will assume that as long as r f <^r v a normal crater is formed
and does not become filled up by ejected material; for r^^ all ejected
material falls back into the crater. Correspondingly, we will have K(z) =
= K x (z) -f K 2 (z) , where K 1 (z)is due to bodies with r'<r t and K 2 (z) to bodies with
^^r^ Relations (31) were written for the former kind of impacts; for the
second integral the upper limit should be r lt The expression for $(h — z)
proves very cumbersome. For p< 4 — 2a it can be approximated satisfactorily
as follows:
(32)
This approximation is justified by the fact that, as will be shown below,
the term T (A'/2) in (26), which depends on \(h — z) (according to (30)), is
small compared with the other terms. In the linear approximation
T(h'l2)=T^f)=T(z)+(^- z y2
and
f(V/2) =*•(«) + £
»i 5
2 S<*-*)
(32 f
Using (26), (30') and (32), we obtain the contribution of bodies between r' and
r' + dr' to #,(z):
^ («)«(*-«)
2 2 £(*-*)
sir'.
(33)
To find K l (z) one must integrate this expression over all bodies producing
craters of depth h^>z, i. e., from r' = z/v x to r 1 — ^.
For KA-iXr,,
I **** = & J Wl-j;W~£ J W^^^Bff^f W (34)
and the ratio of integrals in the right-hand side of (33) is equal to S(fc — z) X
X(4 — p — 2<z)/(5 — p — 2a). Since for r'-^rj £a*0.1^1, the expression given for
the ratio of the integrals is approximately suitable for the entire interval of r' ,
Consequently in view of (27) and (32) we obtain
*iMȣ \ [vf^-z'-Mv'-zjT] v*r"-'dr',
»/*,
(35)
where
(4 — p — 2a)T v«-t
b- P
2°)T^T r 2A7?« 1T-1 = 4-2a
~ 2a L(P-1)vWJ ' T ^^
(36)
173
Integrating we obtain the following expression for z > ^r Q :
*i<«= M 1 -^^ + FlS C2 - o 2 [i -(c 3 C)— -]}, (37)
The coefficient c s inside the last bracket in (37) is slightly greater than
unity. For z<^^r the terms containing z in the expression for K l (z) will be
different, but negligible compared with terms not containing z, which
remain as before. At depths z^/^only the first term is significant. The
factor c 2 in front of the square brackets is small compared with unity. Thus
for p= 3.5, a = 0.15, r = 1 m , A/? = 35 km, r x - 20 km, v, = 4 and v 20 = 18, the
factor c 2 is 0.1. When one allows for the departure of the temperature march
from linearity c 2 may increase slightly, but hardly more than by a factor of
two, and therefore the role of the term associated with T(k'/2)is generally
small.
To evaluate K 2 (z) we will adopt the following simplified scheme to describe
mixing in impacts of the loosening type. As an increase in the size of
falling bodies will not be accompanied by an increase in the rate of ejection
of material, the rate of mixing l m should be bounded. We assume that an
elementary volume lying at a depth z before impact will lie with equal
probability in the depth interval (z — l m , z -f-/ J after impact. Furthermore, l m
should be of the order of the maximum height of lifting of material on impact.
A reasonable value for l m would be / m = v 1 r 1> i. e., the maximum depth from
which material was ejected during formation of the largest crater (before
transition to a loosening explosion).
Reasoning in the same way as for K 1 (z) t we find that the amount of heat
(divided by cp) transported upward together with material across 1 cm 2 at a
depth of z on impact of a body r'is given by
(A-z)7(A+±) for h-z<l m
and
h will be understood to mean the maximum depth from which material is
lifted on impact. We will take, as before, /t = v,r'. When the uplifted
material sinks back, the amount of heat carried down across the same
small area is
(h-z)T(hj2), if h-z<l m% z<l mf
(A _ 2) r(i±4pia), if h- 2< i m , *>/„,
l m T(i) 9 if h-z>l m% z>l m .
5979 174
The resulting flux of heat to the exterior for one impact of scale h is
given by
1 dT
T*- min «*- 1 )'- <*-*)*. v- w.
(38)
where min {...} denotes the smallest of the four products in braces. As in
the derivation of K x (z), we assume* a linear march of temperature with depth
as an approximation.
Summing this expression over impacts of all bodies larger than rjand
dividing it by dTjdz, we obtain K 2 (z);
* 2 w » 4* i min < z (h - 2)j *'- '« (h ~~ 2) ' '-y ** r ' =
= -^T$ min{r(A-2), zl m , l m (h-z), IJJy^'dr'. (39)
Let us take l m = ^r lt and v a = const = v^ f-^Y and set z/v/j — C. Then for C<1
^W = ^F^ J (^-1)^'^ + /,,, J r*"'*' ;
*. (0 - <* [dr (1 + C)4 ~' " c - ^f -feH •
where
°2 8(p-3)r*
Similarly for l<C<r^/2r, we obtain
*• w = c » {1=7 m + v* - c 4 -'] - fay*}
and for r # /2r, < C < r j,/r,
^^^[^(i)--^^^].
(40)
(41)
(42)
(43)
The complete coefficient of thermal conductivity K (z) should also include
the ordinary thermal conductivity k (z) (molecular, radiant, etc.):
*M = jr,(*) + jr t (x)+*(i).
However the role of k (z) becomes significant only at depths z close to \ x r M ,
where the sources of energy £ are insignificant and T(z) practically ceases
to increase with z. Therefore k (z) has little influence on temperature.
To make sure that the value obtained for K (z) is sound, it would be
desirable to evaluate it in some other, independent way. In principle the
process of mixing by impacts from falling bodies is somewhat similar to
the turbulent mixing of fluids. There heat transfer is usually described by
means of the coefficient of "turbulent thermal conductivity" (Landau and
Lifshits, 1953, p. 252):
^turb^" ^turb •
175
where I is the characteristic dimension (usually maximal). This gives K tmb
only up to a constant factor (of the order of one) which is determined experi-
mentally. In our case experimental determination is impossible. On the o
other hand, thermal conduction is essentially a diffusion process. In gases,
for example, the coefficients of thermal conductivity, diffusion and kinematic
viscosity are identical and have the same form as tf tU rb :
ft = /) = v = ii;\ = l^x ) (44)
where X and t are the mean free path and time of the molecules and v is their
mean velocity. In cratering mixing, due to the smallness of the coefficient
of ordinary molecular thermal conductivity, heat transfer takes place mainly
together with transfer of material. The latter is random in character and
can be described by the methods of the theory of random motions.
In the simplest case pf random one- dimensional displacements of a
particle along a straight line, by the same distance I every time (length of
step), with equal probability in both directions and with frequency n, the
mean square displacement of the particle from its initial position within the
time t is given by
V^=y/2Dt, ( 45 )
where D is the coefficient of diffusion, which is given by
D = \nl* (46)
(see Chandrasekhar, 1943). If a particle describes displacements of varying
length I. with frequency n, , then
*>=42»,/?=|i,p.
(47)
where re — 2 rt .*
From this, in particular, it is easy to obtain the expression of D written
above for a gas by noting that
T* = jW = l.l*=l\* and «= 1/x.
Thus in the mixing of material the mean value of the coefficient of thermal
conductivity can be taken in the form (47):
x '=i2^=l* (48)
It should be stressed that / should be interpreted in our case neither as the
absolute displacement of a volume element of material along z, nor as the
variation of its distance from the actual (not mean) surface, which changes
position stepwise in each impact. Since we are interested in mixing in the
sense of temperature equalization, /should be the measure of the temperature
difference between volumes of material being mixed.
176
When a crater of depth A=v 1 r' is formed, most of the material is ejected
to the surface, i. e., for an element at depth z the scale l*~z. A significant
fraction of the material spills on the bottom from the edges of the crater.
For the spilled material l~-h— z. Some of the ejected matter lands at the
bottom of deep craters that in some cases have not cooled down. To some
extent this increases / for small z and decreases it for large z. On the
average it seems one can assume
h
P«XA'»?=lj *& = .*.#. (49)
Integrating over all craters of depth z<A<v 1 r 1 , we obtain
This expression differs from our earlier expression (37) for K x (z) only in a
factor X and in the form of the factor in brackets. As no account is taken in
K' x (z) of ejected material reaching deeper craters and contributing to /*, the
agreement between K i and K[ can be regarded as satisfactory.
In impacts of large bodies with r f >r lf it is presumed that a volume
element can shift upward or downward by any distance less than l m . Then
/ ? = /« l /3 and for zO^
*; W =4 r^=4H^W= i*,[i -fen. (51)
while for v 1 r 1 < z < v t r M
iw=*r« &, =TC.[(^r-ten- (52)
Comparing these expressions with (40) — (43), we see that the differences
between K 2 and K' 2 are also relatively slight. Figure 12 shows the values of K x
and A,, K[, K' t and their sums K = K l + K i and K' = K[-\-K' t as a function of
C = z/v 1 r 1 and corresponding to p= 3.5, r, = 40 km, r M = 100 km, and r = 1 m.
It is interesting to note that the difference between K and K' is considerably
less than that between individual components. The agreement is even
excessive in view of how different the methods used to evaluate K and A"'are,
and how many simplifications they contain. It gives us ground to hope that
the principal features of heat transfer in impact mixing have been elucidated
correctly.
In neither method, however, was allowance made for crater overlap. As
a result of overlap the mixing depth increases while the function K (z) becomes,
as it were, blurred," increasing for small and large z. Sufficiently accurate
allowance for this factor would require cumbersome calculations, on which
we will not dwell here. We merely note that a very approximate estimate
causes K x (0) to roughly double. The corresponding function K (z) used to
calculate terrestrial temperatures for the foregoing values of the parameters
is represented in Figure 12 by a solid line.
177
Another major factor governing the warming of the growing Earth is the
depth distribution of the energy liberated during impacts from falling bodies.
It can be determined by the following
methods:
1. A fraction r\ { of the total energy goes
to the bottom of the crater (more precisely
into the material not thrown out of the
crater). Propagating with the shock wave,
it is expended chiefly in warming and
destroying material. A small part t) # of
this energy goes to great depths in the
form of an elastic seismic wave. This par
amounts to about 1% of the entire impact
energy (Bune, 1956; Pasechnik et al.,
1960; O'Brien, 1960; Kirillov, 1962). We
note that 1% of the Earth's gravitational
energy corresponds to warming of the
Earth by 400°. But estimates of t|, are not
reliable enough.
2. A fraction of the total energy t| # = 1 —
-T| t .is thrown out of the crater together with
the material. Most of it (of the order of r\ t
in quantity) converts into heat, a fraction
f\ m converting into energy of motion of
(apparently small) being expended in evapo
FIGURE 12. Coefficients of thermal conduc-
tivity as determined by different methods: K
— from the expression for the heat flux; A"'—
by analogy with the coefficient of diffusion.
ejected material and a fraction
ration of the material.
The main heat sources of the Earth are the large bodies. The role of the
small bodies reduces chiefly to participation in the heat balance of the
surface layer of the Earth, i.e., to the creation of a definite surface tempe-
rature T^ t It is therefore most important to establish the depth distribution
of the energy liberated in impacts of large bodies.
In the flat one- dimensional problem the simplest form of wave damping is
exponential. It also gives an exponential function for the generation of the
energy E (z) per unit volume
• / e~ h '
dJ
^=E(z) = bE,e~
where E = j E(z)dz is the energy liberated at all depths (in a column of
2 °
1 cm cross section).
For a spherical wave (concentrated impact) in the simplest instance of
damping (constant absorption coefficient) we have
'(')=-£«-*. •<r) = 6-£«r*.
(53)
To obtain the energy E (z) liberated in a unit layer at depth z, one must
integrate e(z) over all r for given constant z. Let
Then
2 = r cos 9, p == r sin 0.
rd8 = cos 8dp
178
and
«/2 bM
E (z) = j e (r) 2n P dp = 2nbJ J e C08fl tg 0d6 = 2*6/ j -
~ x dX
Since all the energy going into the hemisphere is given by E = 2nJ , we have
E(z) = bE Q E l (bz) i (54)
where
00
dx
E l{ x)=\e-*-
The inverse of b } which represents the characteristic wave damping
distance, can be taken to be proportional to r 1 : b=\j$r'. If we estimate b
from the condition that 1% of the entire energy escapes below the zone of
destruction (^2h), we obtain 3^2.
The distribution of the energy liberated in the Earth in the fall of bodies
of various sizes can then be written as
B(*) = -%
(55)
where v Q is the impact velocity, assumed to be the same for all bodies; h x is
the thickness of the layer produced by material thrown out of the crater
(see (14)), r } is the body radius at which "loosening" impacts set in (with
nearly all the material falling back into the crater); pj^p 3 ^2, p 2 »*l. The
fraction of energy t\„ expended in evaporation does not exceed a few percents
and will therefore be disregarded. The second integral is very cumbersome.
For purposes of an approximate numerical estimate of E{z), it is desirable
to simplify it. Since for r'<^r 1 the layer thickness hj is very small compared
with r' t the heat liberated inside this layer is almost entirely emitted from
the surface, without contributing to the warming of the Earth's interior.
Only for r' approaching r t does the second integral become comparable with
the first. Let us therefore take
E(z)
4fi
L r tn
(it + 7 i < r 7^i)w f
s *(^0 B(i
,r')dr> +
for ] VlV / V f
(56)
If n(r') gives the number of bodies of radius r' falling per second over the
entire Earth, E{z) should also refer to the whole Earth. Given a power law
of distribution of the falling bodies n (r l ) = CV"' , the energy liberated per unit
volume is given by
«(<) =
E(z) _ vl C't'
47tr2
3r2
^ + ^7^
^(^y^'+i-k^^-y^'
(57)
179
where r is the Earth's radius. Quantity C is determined from (28). For
p<4 the quantity £ in the equation of thermal conduction (51) is given by
'w-^-^^f^f fc+vWH*** J rat+£ jV'*' j r»&l (5 8)
where c is the heat capacity of the material of the Earth. We will take
rt 1 = const, t) # = const, and p 1 = p 3 = [J.
By changing the order of integration the double integrals can be reduced
to single integrals. We set z$r ( = x, z$r m = x m and so forth. Then
<-m X Xi */?X X,,! r m
If J
■p-3
f^wj^a= r L_UB l ( Xl )-^E lfeJ -(^'^|
and
I V L X^ Xi J
(59)
In particular, for p= 3.5
("'-'■■?')l/? E ' < '- > - E '(»'l <60)
Since p <^ 4, one can take r m = and x = °° • Then for p = 3.5
£(*) =
2pcr^
[(l~7 1( _27,, Xl )erfv6a + L +
+ 2^X l -erfV^7J-2T ]- y / ^6^+ (1 - % + n t ) |/^ E, ( Xl ) - E, ( Xj ,)J. (61)
The functions tf(z) and <£(z) which we obtained permit us to calculate Earth
warming due to impacts from large bodies (first term in (12)). Unfortunately
there are not enough data to allow us to dwell on definite values of the
180
parameters in the formulas for K(z) and £(z). A calculation carried out for
p= 3.5, r x = 40 km, r M = 100 km, r\ { = 0.4 and \ = 0.6 indicates that warming
due to large bodies amounts to about 1100° at a depth of 400 — 500 km below
the Earth's surface. Warming increases with increasing r M and decreases
with increasing r, and p.
The maximum warming at depths of
400 — 500 km due to impacts of large
bodies will be denoted by T M . From (61)
it is apparent that the energy liberated
in impacts is proportional to ujccm 1 /'.
For smaller m, moreover, sizes of
falling bodies were also smaller. The
liberated energy penetrated to lesser
depths and less of it remained inside the
Earth. Body sizes r M can be assumed
to be proportional to rccm ,/a > while their
depth of penetration w oc r M v^ cc nC 1 *
(Andriankin and Stepanov, 1963). One
can assume approximately that the
warming of the Earth due to body impacts
was proportional to m u , where u
apparently lies between 1 and 2. Then
from (13.12) the distribution of the
primary temperature inside the Earth
can be written as
fm/q) f >
FIGURE 13. Primary temperature of the Earth:
T 9o — warming of Earth during growth process
by impacts from small bodies and particles;
Tmin — corresponding initial Earth tempera-
ture with allowance for contraction and radio-
active heating during growth process (100 mil-
lion years); T $ — warming of Earth by impacts
of bodies of various size, including large ones;
r— corresponding initial Earth temperature.
T{m)=[T*+T K (jff]A{m) + B{m)*(t-tJ,
(62)
where T si) is the warming of the Earth
due to impacts of small bodies and
particles, determined according to (62).
We noted earlier that calculations
point to values of about one thousand
degrees or possibly slightly more. Assuming T x = 1000° and u = 1, we
obtain the distribution of the primary terrestrial temperature T given in
Figure 13. For comparison we give the curve of the temperature T a0 ,
obtained under the assumption that all the material landing on Earth during
its formation consisted of small bodies and particles, and the corresponding
curve of the initial temperature T miu . We see that the maximum initial
temperature occurred in the region of the upper mantle and may have
exceeded 1500°K. For T M = 1200° it is nearly 2000°K. The importance of
this preliminary result for the study of the Earth ! s thermal history makes
it imperative that the parameters appearing in the foregoing formulas and
governing the initial temperature of the Earth be made more accurate.
181
Chapter 16
PRIMARY INHOMOGENEITIES OF THE EARTH'S MANTLE
41. Inhomogeneities due to differences in chemical
composition between large bodies
A number of recent, independent data indicate that there are pronounced
horizontal inhomogeneities varying in scale and extending to various depths
inside the Earth's mantle. Gravimetric maps clearly show positive and
negative gravity anomalies covering areas several thousands of kilometers
in diameter (Lyustikh, 1954). Analysis of zonal harmonics in the Earth's
gravitational potential detected by satellite measurements (O'Keefe et al.,
1959; King-Hele, 1962) has revealed (Munk and MacDonald, 1960;
Mac Donald, 1962) that the observed anomalies are considerably greater than
the anomalies calculated for the continents under the assumption of hydro-
static equilibrium (isostasy), and are opposite in sign. They could not be
due to density variations in the crust and are undoubtedly caused by large-
scale horizontal inhomogeneities in the Earth's mantle. Studies of tidal
deformations of the crust (Parijsky, 1963) show that the elastic properties
of the mantle in the European sector of the USSR and in Central Asia are
different. Electromagnetic and seismic observations also point to the
existence of regional inhomogeneities in the Earth's mantle (Tikhonov etal.,
1964; Fedotov and Kuzin, 1963).
The presence of the oceans and continents is also evidence of large
horizontal inhomogeneities. It is hardly likely that such large formations
could have developed in a primordially quasihomogeneous Earth. Their
existence is nather to be viewed as evidence that the Earth's mantle contained
large-scale pnmary inhomogeneities.
Evidence indicating the probable existence of large-scale primary
inhomogeneities inside the mantle is also provided by the study of the Earth's
formation. We noted in Chapter 8 that large bodies must have constituted a
considerable fraction of the mass of solid material from which the Earth
was formed. In Chapters 9 and 11 it was shown that the masses of the
largest bodies falling on the Earth were of the order of one thousandth of the
Earth's mass. A striking illustration of the important role of large bodies
in the formation of the planets and their satellites is provided by the lunar
craters and seas. The lunar seas were formed as a result of planetesimals
a few tens of kilometers in diameter striking the Moon. Additional masses
(mascons) inside them detected recently from gravity anomalies point to
considerable inhomogeneity of the Moon's outer layers. Many large craters
have also been discovered on Mars (Leighton et al., 1965).
182
There are two possible types of inhomogeneity traceable to large bodies
striking the Earth: inhomogeneities arising from differences in the chemical
composition of incoming bodies, and inhomogeneities due to impacts
accompanied by liberation of large amounts of energy (Safronov, 1964 b;
1965b). Let us begin with the former.
The chemical composition of the planets varies regularly with distance
from the Sun. The inner planets are denser than the outer ones. Urey,
Elsasser and Rochester (1959) record density differences of up to 0.2 g/cm 3
in stony meteorites, due probably to the fact that their parent bodies
originated in different regions of the asteroid zone. Variations have also
been detected in the composition of iron meteorites. At standard pressure
Mercury has the densest material in the solar system. The density of Venus
is apparently several percents higher than that of the Earth (Kozlovskaya,
1966). The source zone of the newly developed Earth extended nearly from
Venus' orbit to that of Mars. One might expect that bodies formed in
different parts of this broad zone would display variations of several
percents in composition and density.
If the bodies were incorporated in the Earth roughly in their original
form as local inclusions, they must have introduced pronounced inhomo-
geneities in the mantle and given rise to motions inside it. Given a resis-
tance threshold of 10 to 10 2 bar for the mantle material, inclusions several
tens of kilometers in diameter having a density that differed from that of
the surrounding material by 0.1 g/cm 3 must have sunk or floated under the
influence of gravitational forces. Larger inclusions could have shifted
starting from smaller density differentials. For sufficiently large inclusions,
differentiation could have begun directly after formation of the Earth.
Smaller inclusions would have begun to shift only after some degree of
warming had taken place and the viscosity of the surrounding mantle
material had decreased.
Differences in the composition of the bodies may have been reflected in
differences in the content of radioactive elements (again amounting to a few
percents), as well as in density variations. The two may have coexisted,
but unlike the latter, the former did not manifest themselves immediately.
Inclusions containing an excess of radioactive elements warmed up somewhat
more rapidly in the process of decay, gradually becoming less dense than
the surrounding material. In these regions partial melting of silicates, their
uplift and the formation of the crust must have begun earlier. Magnitskii
(1960) has attempted to explain existing gravity anomalies with the help of
inclusions several hundreds of kilometers in diameter and having an excess
of radioactive elements.
As has already been mentioned, these inhomogeneities associated with
variations in the bodies' chemical composition would have occurred if the
bodies became incorporated in the Earth in the form of local inclusions and
did not disperse on striking the Earth. The velocities of the falling bodies
were only slightly in excess of the parabolic velocity ( v = v t \j\ + 1/26 ), but
at the closing stage of the Earth's growth they nevertheless amounted to
10— 12 km/sec. Impacts of such velocity would lead to disintegration of the
bodies and scattering of their material over a large area together with the
material thrown out of the crater. This means that inhomogeneities associa-
ted with differences in composition must have been largely smoothed over.
183
In experimental studies of a tinted droplet in a container of water, the
falling droplet spreads out over the surface of the "crater". In certain cases
the "crater" collapses and a cumulative jet ending in a droplet containing
almost all the water of the original tinted droplet rises upward from its
center (Charters, 1960). Attempts have been made to attribute the presence
of central mounds inside many lunar craters to a similar sliding of material
from the edges of the crater toward the center (Shoemaker, 1962). If the
central mounds really consisted to a large extent of fragments of a fallen
body, the inhomogeneities introduced by the bodies should be fairly evident.
But the brittleness of rocks makes them very different from liquids and
metals. Charters notes that no fragments of the projectile were found at the
center of the crater in experiments with rocks. It seems that bodies of
silicate composition scatter on falling at high velocities.
However, the situation changes drastically in the case of impacts of large
bodies. It was shown in Chapter 14 that when very large bodies fall gravita-
tion becomes important and geometric similitude is destroyed. Qualitatively
the picture is as follows. As the size of the falling bodies increases, the
depth of penetration ("explosion" depth) increases accordingly. But since
the energy per unit mass does not change with increasing size (same rate of
fall), the initial scatter velocity of the material and correspondingly the
scatter distance remain as before in the first approximation. Therefore for
sufficiently large body sizes the radius of the crater formed in the case of
geometric similitude, would exceed the distance of scatter of material from
the crater. As a result nearly all the ejected material would fall back into
the crater and all the energy liberated on impact would be trapped together
with the material inside the crater. The fall of such large bodies is similar
to camouflets or loosening explosions. The material of the falling body is
not scattered in this case, but remains within a closed volume exceeding the
volume of the body itself by no more than one order. A deviation of a few
percents on the part of the chemical composition of the body from that of the
Earth will cause a volume equal to several body volumes to deviate in
composition by tenths of a percent and correspondingly to deviate in density
from the mean density of Earth material by -0.01 g/cm 3 . Such an inclusion
will cross the resistance threshold and sink (or float up) if the body producing
it measures about 100 km in diameter or more.
42. Inhomogeneities due to impacts of falling bodies
As a result of the disintegration of the material adjacent to the crater by
the shock wave, and also due to ejected material falling back, a layer of
crushed rock (breccia) is formed under the crater. Data obtained by drilling
in the Holliford and Brent craters agree with Rottenberg's theoretical
conjecture that in granite gneisses the depth of the breccia in the central
portion of a crater should amount to about one third of the crater diameter
(Beals et al., 1960; Innes, 1961). In the disintegration zone the pressure
along the front of the shock wave was greater than the rocks' resistance
and oscillations propagated inelastically. In the first approximation the
volume of crushed rock is proportional to the total impact energy of the body
(Innes, 1961; Baldwin, 1963). The density of material in the breccia region
is lower than in adjacent regions where the material was not subjected to
184
disintegration. According to drilling data, confirmed by independent
estimates based on the measured values of gravity anomalies, the density
difference averages 0.2 g/ l 3 (Innes, 1961). The volume of the breccia is
several times greater than that of the crater and therefore much greater
than that of the fallen body. Density inhomogeneities stemming from the
disintegration of material in impacts are greater than those discussed in
Section 41. The foregoing data, however, refer to depths of the order of
1 km. At depths of interest to us (of the order of hundreds of kilometers)
the density drop after impact should obviously be substantially less, and
its neutralization should proceed more rapidly.
More definite statements can be made regarding thermal inhomogeneities
produced in the fall of large bodies. A considerable fraction of the shock
wave energy converts into heat in the breccia zone. Opik (1958) worked out
an approximate scheme for the impact mechanism which he used to evaluate
warming up of material inside the impact zone. For a rate of fall of
10 km/sec, this warming amounts to 580°, 208° and 93° along the frontal
surface of a wave occupying, respectively, a 30- fold, 50- fold and 7 5- fold
mass of the falling body. Although this estimate is apparently somewhat
exaggerated, it shows that a mass of material considerably greater than the
mass of the body itself will warm up by hundreds of degrees. The thickness
of the warmed layer is of the order of the diameter of the fallen body.
Experimental data show broad differences between different rocks as regards
warming on impact (Chao, 1967). Thus quartz requires a peak pressure of
400 kbar to warm up by 600°, while sandstone needs only 90 kbar. The
pressures necessary for warming by 1500° are respectively about 500 and
200 kbar.
Only the largest of the bodies could have caused significant temperature
inhomogeneities. First, for any reasonable distribution function the number
of bodies decreases with increasing size. Therefore the deviation of any
random quantity from its mean value of l:\JN will increase. Second, it is
only when very large bodies are involved that nearly all the energy liberated
on impact will remain in the impact region. Third, only the largest bodies
were capable of warming a layer so thick that the lower portion of the layer
was situated below the mixing region due to impacts of other bodies and was
not subjected to the effective cooling associated with this mixing. Fourth,
the larger the fallen body, the larger the region heated and the greater the
time required for the temperature of this region to level down to that of the
surrounding medium.
An idea of the body sizes capable of giving rise to long-lived thermal
inhomogeneities can be formed by estimating the rate of cooling of a plane
layer near the Earth's surface (Safronov, 1965b). For the initial and
boundary conditions
r <*.0>=( x>h (1)
r(0, o = o
the solution of the equation of thermal conduction without sources for a plane
half- space has the form
T(x t = ^-[2erf(y)-erf(y-j/)-erf(i/-hj/)], (2)
185
where y = xl2\fki and y l =hj2\Jki. The temperature at the middle of the layer
x = h/9 i« cfi\re*n hw
x = h/2 is given by
•= r (T-0=^[MT)— -W)]. ^
Below we give the ratio TJT X for various layer thicknesses h, 10 9 years
after cooling begins, for k= 0.01:
A > km 50 100 200 300 500
T d T l 0.002 0.012 0.080 0.21 0.52
At the center of a layer 300 km thick, one billion years later about 20%
of the of iginal temperature excess is still left (i. e., about 100° if the layer
was warmed by 500°). Consequently bodies several hundreds of kilometers
in diameter must have induced considerable thermal inhomogeneities in the
developing Earth while falling. Larger inhomogeneities lasted 1 — 2 billion
years, i.e., until such time as intensive warming by radioactive heat had
caused the viscosity of the Earth's material to alter appreciably, with
gravitational differentiation setting in. In sections of the upper mantle with
temperature excesses of 100 — 150° stemming from impacts of large bodies,
the crust material must have begun to melt 100 million years earlier than
in other zones.
Recently it has emerged that the energy of lunar tides inside the solid
Earth may have dissipated preferentially in warmer sections of the upper
mantle with material of low elasticity, imparting to these sections a
distinct additional warming (Ruskol, 1965). Thanks to this energy source
thermal inhomogeneities could have survived over longer periods, possibly
even increasing in intensity. The impact areas of the largest bodies striking
the Earth during its formation could have converted into relatively stable
regions of higher temperature in which all processes associated with melting
of the crust and tectonic activity began earlier and proceeded with greater
intensity. These regions could have remained active for a long time, until
such time as they had lost a considerable fraction of their radioactive
elements by migration of the latter to the surface together with the light
melts which produced the crust.
The most important inhomogeneities must have been due to bodies several
hundreds of kilometers in diameter; their dimensions must have been in
excess of one thousand kilometers. This fact permits us to conjecture that
there may have been a connection between initial thermal inhomogeneities in
the mantle (due to impacts of the largest bodies falling on the Earth) and the
subsequent differences in the thermal history of these regions which led to
the formation of the continents. Though many factors influenced the complex
process of the Earth's evolution, it would seem that initial thermal inhomo-
geneities played a particularly important role and must be taken into account
in any study of this process.
186
CONCLUSIONS
The theory of planet formation by accumulation of solid bodies and
particles has provided the Earth sciences with important information about
the initial state of the Earth and especially about its primary temperature.
If all gravitational energy liberated in the Earth's formation had remained
in its interior, the Earth would have warmed up to 40,000°. Therefore
estimates of the primary temperature will depend essentially on how the
Earth is assumed to have been formed. The old view that the planets were
formed by the condensation of gaseous clusters, and that the Earth was
originally in a hot, molten state, has long held sway. Schmidt's conversion
to the idea of planet formation by fusion of solid bodies and particles led
him to draw the important conclusion that the Earth was initially in a
relatively cool state, a conclusion that had an important influence on the
subsequent development of the Earth sciences. Given this method of
formation, the main sources of heat during the period of the Earth's growth
must have been impacts of falling bodies, the contraction, of its material
under the pressure of layers accumulating at the top, and the generation
of radioactive heat. The warming resulting from contraction is proportional
to the temperature of the material being compressed. In the mantle at the
boundary of the core, contraction caused the initial temperature to increase
1.9 times; at the Earth's center (for a metallized silicate core) it caused an
increase of 2.1 times, assuming that phase transition takes place without
generation of heat. The total energy of contraction of the Earth amounts to
4.2 * 10 38 erg, of which 2.3 * 10 38 erg is expended in the phase transition. The
conversion into heal of just one tenth of the transition energy would warm
the core by 1000°. It would therefore be important to have an estimate of
the amount of energy converting into heat in the phase transition.
Owing to the comparatively short time within which the Earth was formed
(10 8 years), warming due to generation of radioactive heat over this period
was limited: the inner part of the mantle was warmed 100°, the core
150 — 200°.
The main heat source of the growing Earth was the impacts of the bodies
and particles from which it was formed. Estimates of the initial tempera-
ture depend largely on the body sizes assumed. The first calculations,
based on the assumption that these bodies were small, produced a low initial
temperature— from 300°K near the surface to 800 — 900°K at the center.
The impact energy of small bodies and particles was liberated near the
surface of the growing Earth, practically all of it being emitted into space.
Even in the period of most intensive growth, the temperature of its surface
would not have exceeded 350 — 400°K.
187
The importance of large bodies in the process of planet formation has
recently begun to emerge (see Part II). The largest bodies falling on the
Earth had diameters reaching several hundreds of kilometers. The larger
the incident body, the greater the depth at which its impact energy was
released and hence the greater the fraction of this energy trapped inside the
Earth, unable to escape into space. On the other hand, larger bodies
produced deeper craters, inducing more intensive mixing of the material on
impact. Heat transfer by mixing of material during the impact of large
bodies is far more efficient than heat transfer by ordinary thermal conduction.
To evaluate the warming up of the growing Earth (i. e., an Earth with a
mobile boundary) due to impacts of falling bodies using the equation of
thermal conduction, it is necessary to determine the analog of the coefficient
of thermal conductivity K and the depth distribution of the energy S liberated
on impact. But evaluation of K and g requires further development of the
theory of cratering, especially as regards the consequences of the fall of
very large bodies (which contributed most to the Earth's primary tempera-
ture). Here the Earth's gravity essentially destroys geometric similitude,
leading to qualitatively new phenomena.
The velocity of the bodies at the instant of impact did not depend on their
size and at the concluding phase of growth amounted to 10 — 12 km/sec. The
depth of penetration ("explosion" depth) is proportional in this case to the
size of the incident body (about twice its diameter). The rate of ejection and
therefore the distance of scattering of the ejected material do not depend on
the body size. In the case of very large falling bodies, the scattering
distance is less than the radius of the crater which would have formed for
geometric similitude. Therefore the greater part of the ejected material
falls back into the crater. Such impacts are similar to "loosening
explosions." As the size of the falling bodies increases, the character of
the crater alters in the same way as when the relative depth of explosion
increases. Although the mass of material over the explosion center per unit
mass of incident body remains constant, the energy required to eject a unit
mass from the crater.inside the Earth's gravitational field increases with
the size of the crater. The relative explosion depth should therefore be
measured not .by the ratio w?/C v ' but by the ratio wfC 1 ^ , where ju is close to 4
(for large bodies). The impacts of bodies with diameters exceeding one
hundred kilometers are similar to loosening explosions. These are the
impacts that warmed the Earth most efficiently, as nearly all the heat
released when they fall remains buried inside the filled- in crater.
Numerical estimates show that the maximum of initial temperature of the
Earth occurred in the region of the upper mantle and probably exceeded
1500°K. This means that the time required for the subsequent warming of
the hotter regions to the melting point of low- melting substances and for
initiation of the process of crust formation may have been short (less than
one billion years). For a more exact estimate of the original temperature
additional research is necessary on the theory of cratering (especially for
large^body impacts) and on the size distribution function of the bodies from
which the Earth was formed. Also required is the construction of a more
rigorous theory of heat transfer in terrestrial material mixed by the
impacts of falling bodies.
The largest of the bodies striking the Earth induces primary inhomo-
geneities in its mantle. One type of inhomogeneity was related to differences
in chemical composition between large bodies. The bodies were formed
188
inside a broad zone between the orbits of Venus and Mars. The density and
content of major chemical elements of these planets differ from those of the
Earth by a few percents. The same order of differences in composition
can be expected to prevail between individual bodies landing on the Earth.
Small bodies scattered over a large area on falling. Very large bodies
intermingle with a volume of material exceeding their own volume by only
one order. Such inclusions could have deviated from the mean density by
-^0.01 g/cm 3 for an initial deviation in body composition of several percents.
For body diameters of over 100 km, such inclusions would overcome the
resistance threshold of the terrestrial rocks and begin to sink or float
(depending on the sign of the density deviation).
Another type of primary inhomogeneity, temperature inhomogeneities,
was due to impacts of falling bodies. Unlike those just discussed, they
could have been produced by large bodies of all compositions. Only the
largest gave rise to significant inhomogeneities. The layer they warmed
was so thick that its lower portion lay outside the zone of mixing by impacts
of other bodies. Equalization of temperature proceeded slowly, and the
inhomogeneities may have lasted 1—2 billion years. Preferential dissipation
of the energy of lunar tides in these warmer regions could have converted
them into relatively stable regions of higher temperature in which processes
associated with crystal melting set in earlier and proceeded more intensively.
These regions had dimensions in excess of one thousand kilometers, and it
is natural to conjecture that the formation of the continents was related to
their presence.
Thus while the theory of planet formation by accumulation of solid bodies
and particles can furnish information of importance for the Earth sciences
regarding the Earth's initial state, to obtain reliable results in this limited
field calls for cooperation of scientists in a variety of specialities. The
early history of the Earth still contains questions relating to the primary
atmosphere, primary hydrosphere, and so on. The study of the initial state
and evolution of the Earth is one of the most pressing problems of geophysics.
Recently initiated comparative studies of the structure, composition and
thermal history of the Earth and other planets may prove of great help in its
solution.
189
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202
SUBJECT INDEX
Accretion 4, 67, 114 144
Accumulation 2, 4, 10, 90ff, 123, 154
of Earth-group planets 105 — 112,
197, 189
of giant planets 136 — 143
Age of Earth 110
of Sun 110
Alfven velocity 9, 51
Angular momentum 6, 7, 11
distribution in solar system 6
transfer from gas to particles 15
transfer from Sun to cloud 6, 9
Anomalies, gravity 182, 183, 185
Asteroids 126, 146— 154
formation 146—148, 153
fragmentation 126, 147, 153
rotation 126, 127, 148—151
C
Capture
by planet of material in satellite
cluster 124
theory 4 — 6
Chemical composition
of planets 6, 26, 138, 183, 184
Coagulation 90 — 104
asymptotic solutions 97 — 104
equation 91, 100
Coefficient of viscosity 176
absorption 35
diffusion 176
Gruneisen 155 — 157
of thermal conductivity 164, 171,
175- 176, 188
of turbulent thermal conductivity 175
Comets, clouds 137, 142, 153'
Condensation of volatile substances 44
Condensations
dust 57, 111, 113, 114
evolution 61 — 66
formation 57 — 61
gaseous 9, 113, 139
mass and size 57 — 59
primary contraction 59 — 61
rotation 59, 61
Continents 182, 186, 189
Convection 8, 9, 17— 19, 67, 161
Cosmic rays 10, 32
Craters 166
central mounds 184
depth of disintegration zone 184
ejection index 168
explosion 167— 170
explosion depth 169, 188
geometric similitude 169, 170, 184,
188
impact 166 — 171
loosening impacts 170, 171, 172,
174, 188
mass of ejected material 167
on Moon and on Mars 182
size 166, 169
Criterion, Jeans 45
Rayleigh 16, 17, 23, 24
Critical density 30, 50, 51
203
D
Damping of turbulence 20 — 22, 26, 67
Deficit in noble gases on Earth 10
Density
critical 30, 50, 51
Roche 28, 30, 57, 66
surface 8, 10, 41, 43, 57, 109,
110, 148, 153
Differentiation of material in Earth 161,
183
Dispersion of velocities of bodies 69 ff,
152
moving in a gas 81 — 82
of equal mass 69 ff
of varying mass 82 ff
Dissipation
of energy of lunar tides 186, 189
of gas from solar system 10, 144,
145
thermal 145, 153
of solar nebula 8
Dissociative equilibrium 150, 151
Distribution
of density in protoplanetary cloud
25
of energy released in impacts 177 —
181
of initial temperature in Earth 181
of masses 96, 97, 103, 108, 152
on fragmentation 102 — 104
of temperature in dust layer 34, 43
Dust layer
disintegration 57, 67
formation 25 — 27
temperature distribution 32 — 36
thickness 27, 29
growth time llO — 111, 153, 158
primary temperature 155—156, 162,
166, 181
rotation 126 — 128
Effect, Poynting — Robertson 6, 11 — 14,
19
Ejection
index 168— 171
of bodies from solar system 136 —143
Equation
of thermal conduction 165, 166
Poisson 45 — 48
Equilibrium
local thermodynamic 33
of rotating systems 47, 50
Fragmentation of bodies in collisions 100,
107, 126, 147, 153
mass distribution of fragments 103,
104
Giant planets 136 — 145
composition 138
ejection of bodies 138—142, 152
growth time 136—138, 152
Gravitational instability 9, 25, 30, 31,
45 ff, 67
in dust layer 27, 85 112
in gas 9, 25, 114, 139
Gravitational paradox 45
I
Earth
age 110
contraction energy 160
core 155, 161
deformation energy 160
Inclinations of planetary axes of rotation
106, 129, 133, 154
Instability
convective 9, 16—19
gravitational 45, 56
near strongly compressed spheroid 23
204
of infinite cylinder 54, 55, 58
of infinite medium 45
of protoplanetary cloud 57, 67, 138,
139
of rotating systems 46, 47
of solar nebula 8, 9
rotational 6, 9, 114
secular 9
Mass of protoplanetary cloud 7, 9, 10—11,
138—140, 153
of dust condensations 57, 58
Meteorites 2, 3, 10, 146, 148
Moon
craters 182
formation of satellite cluster 123
initial distance from Earth 126, 128
J
N
Joint formation of Sun and cloud 6, 67
theories 6—11
Jupiter
accretion of gas 138, 144
chemical composition 6
ejection of bodies into zones of
other planets 137— 140
from solar system 137 — 143
initial temperature 138
Largest bodies falling on planets 106,
129—135
Law of planetary distances 107, 153
Libration points 107, 124, 143
Linearized theory of instability 46, 51
M
Magnetic field 5, 6 — 9
influence on cloud' s stability 23
terrestrial 161
transfer of angular momentum 6,
7, 9, 14
Mantle, Earth 1 s
horizontal inhomogeneities 182
initial temperature 181
primary inhomogeneities 182— 186,
188
Mars
craters 182
slowing down of growth 111, 112,
148
Neptune 126, 134, 136—145
growth time 136— 138
violation of Bode* s law 140
Number, Reynolds 16, 28, 29
Planet embryos 83, 85, 88, 105 — 108,
147, 152
Planetesimals 113, 183
Protoplanetary cloud 2, 3, 16, 25, 32,
47, 67, 138ff
formation 4ff
Protosun 6, 7
R
Relaxation
of stresses 160
time 72, 80, 83, 85
Roche boundary 28
Rotation
empirical dependence 126— 128
inclinations of axes 106, 129 — 135
of asteroids 127, 148 — 151
of dust condensations 60, 61
of Earth 126 — 128
of planets 113— 128, 154
of Uranus 133— 135, 154
Scattering of light
isotropic 34, 36, 37, 42
205
Rayleigh 42, 43
Seismic energy in impacts 177
Solar activity 7, 8, 10, 32
Solar nebula 6, 7
Stability of circular orbits 16
Sun
formation 5, 10
rotation 5, 7. 11
Thermal inhomogeneities in the Earth 1 s
mantle 185, 186, 188
Tides 113, 126, 186, 188
Triton 125
Turbulence 8, 9, 10,
damping 20, 21, 22, 26, 67
scale 30 #
transport of material and angular
momentum 8, 20
Temperature
of Earth, primary 155—160, 162,
166, 181
of Jupiter and Saturn 160
of protoplanetary cloud 32 — 43
Theory
Alfven 5
Cameron 8, 9
Hoyle 6—8, 11
Laplace 4, 113
Lyttleton 5
Schatzman 9—10
Shmidt 4
Thermal conductivity
coefficient 164, 167, 175—176
U
Uniform thickness 25
of cluster of bodies 109
of dust layer 29, 30
of gaseous component of cloud 29,42
Uranus 136— 140
inclination of axis of rotation 134 —
135, 153
W
Warming of Earth
by impacts of falling bodies 161 —166,
180, 181, 185, 187, 188
by radioactive heat 155 — 160, 187
due to contraction 155 — 160, 187
5979
206
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